#6388235

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 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)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22)(23,27)(24,34)(25,37)(26,29)(28,35)(30,39)(31,45)(32,41)(33,49)(36,44)(38,43)(40,47)(42,48)(46,52)(50,51);(1,3,4,2)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23)(22,27)(24,31)(25,33)(26,32)(28,40)(29,41)(30,42)(34,45)(35,47)(36,46)(37,49)(38,50)(39,48)(43,51)(44,52);(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,26,36)(22,28,29,43)(23,31,32,46)(25,38,39,35)(27,40,41,51)(30,44,37,34)(33,50,48,47)(42,52,49,45);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(22,29)(23,32)(28,43)(30,37)(31,46)(33,48)(34,44)(47,50);(6,14)(11,15)(12,13)(19,20)(23,32)(27,41)(31,46)(33,48)(40,51)(42,49)(45,52)(47,50)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=dfi,e^b=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6295431);