#146453

order	512
nilpotency class	4
isoclinism	isoclinic class of #13068
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1893
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10721
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

861 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=di,e^c=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314220
maximal normal subgroup of	#72314220

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53649);