#6777008

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1279 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)(13,18)(14,17)(15,21)(20,24)(22,23)(25,26)(27,37)(28,42)(29,35)(30,36)(31,47)(32,41)(33,43)(34,44)(38,53)(39,49)(40,51)(45,50)(46,52)(48,55)(54,56);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24)(25,27,30,41)(26,32,36,37)(28,38,44,39)(29,45,31,40)(33,48,52,54)(34,53,42,49)(35,51,47,50)(43,56,46,55);(2,5)(6,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,38)(29,43)(30,44)(31,46)(32,48)(34,35)(36,52)(37,54)(39,41)(40,55)(42,47)(45,56)(49,50)(51,53);(1,3,4,7)(2,6,5,8)(10,17)(15,19)(16,18)(23,24)(25,28)(26,34)(27,39)(29,46)(30,44)(31,43)(32,49)(33,35)(36,42)(37,53)(38,41)(40,55)(45,56)(47,52)(48,50)(51,54);(2,5)(6,8)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(25,29,30,31)(26,35,36,47)(27,40,41,45)(28,43,44,46)(32,50,37,51)(33,34,52,42)(38,55,39,56)(48,49,54,53)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=h,e^2=f,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, 6684204);