#6399016

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=efi,g^a=gi,d^b=df,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6306212);