#6875701

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=fi,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,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, 6782897);