#146434

order	512
nilpotency class	4
isoclinism	isoclinic class of #126701
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1735
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 ◁ C₂²×M₁₆ ◁ 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)(3,13)(4,19,21,9)(5,23)(6,27)(7,11)(8,15)(10,25)(12,24)(14,38,40,35)(16,29)(17,46)(18,30)(20,50,52,54)(22,33,51,49)(26,55)(28,39,47,41)(31,45)(32,53,48,64)(34,56)(36,44)(37,43)(42,59,61,60)(57,63,58,62);(1,3,5,16,24,43,56,44,7,18,26,15,6,17,25,45)(2,8,10,30,27,36,55,37,11,29,34,13,12,31,23,46)(4,14,20,41,51,61,64,62,21,40,52,39,22,42,53,63)(9,28,32,38,49,58,50,59,19,47,48,35,33,57,54,60);(1,4)(2,9)(3,14)(5,20)(6,22)(7,21)(8,28)(10,32)(11,19)(12,33)(13,35)(15,39)(16,41)(17,42)(18,40)(23,54)(24,51)(25,53)(26,52)(27,49)(29,47)(30,38)(31,57)(34,48)(36,58)(37,59)(43,61)(44,62)(45,63)(46,60)(50,55)(56,64)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307242, #72307247, #72307259, …
maximal normal subgroup of	#72307242, #72307247, #72307259, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53630);