#7622528

order	512
nilpotency class	4
isoclinism	isoclinic class of #33404
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#106
commutator	C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂	C₂³×#106
prime radical	C₂²×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1865 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,14,18,38)(4,10)(5,11)(6,25)(7,28)(8,13)(9,17,31,44)(12,26)(15,36,39,57)(16,37,19,46)(20,32)(21,33)(22,49)(23,51)(24,35)(27,52)(29,41,54,59)(30,43,53,45)(34,50)(40,58,42,62)(47,63)(48,56)(55,60,64,61);(1,3,6,17,7,19,26,45,8,18,27,44,5,16,24,43)(2,9,12,14,11,30,25,37,13,31,35,38,28,53,52,46)(4,15,22,41,23,42,50,61,20,39,47,59,21,40,48,60)(10,29,34,36,33,55,49,58,32,54,56,57,51,64,63,62);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,20)(9,29)(11,33)(12,34)(13,32)(14,36)(16,40)(17,41)(18,39)(19,42)(24,48)(25,49)(26,50)(27,47)(28,51)(30,55)(31,54)(35,56)(37,58)(38,57)(43,60)(44,59)(45,61)(46,62)(52,63)(53,64);(4,20)(10,32)(15,39)(21,23)(22,47)(29,54)(33,51)(34,56)(36,57)(40,42)(41,59)(48,50)(49,63)(55,64)(58,62)(60,61);(1,5,8,7)(2,11,13,28)(3,16,18,19)(4,21,20,23)(6,24,27,26)(9,30,31,53)(10,33,32,51)(12,25,35,52)(14,37,38,46)(15,40,39,42)(17,43,44,45)(22,48,47,50)(29,55,54,64)(34,49,56,63)(36,58,57,62)(41,60,59,61)

Polycyclic presentation:

2^9|a^2=g,b^2=f,e^2=i,f^2=h,h^2=i,b^a=bf,f^a=fh,h^a=hi,g^b=gi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285601, #72286027, #72286071, …
maximal normal subgroup of	#72285601, #72286027, #72286071, …
maximal quotient of	D₄×#2155, C₂×#7622528, #72320518, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529724);