#6807737

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3204	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1815 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fh,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6714933);