#6802969

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1265 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)(25,26,31,33,30,36,29,35)(27,32,42,47,37,46,39,44)(28,34,43,48,40,49,41,50)(38,45,53,56,51,55,52,54);(1,2,4,5)(3,7,8,6)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24)(25,27)(26,32)(28,38)(29,39)(30,37)(31,42)(33,47)(34,45)(35,44)(36,46)(40,51)(41,52)(43,53)(48,56)(49,55)(50,54);(2,5)(6,7)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(26,33)(27,37)(29,31)(32,44)(34,48)(35,36)(38,51)(41,43)(45,54)(46,47)(49,50)(55,56);(10,17)(15,19)(16,18)(23,24)(25,28)(26,34)(27,38)(29,41)(30,40)(31,43)(32,45)(33,48)(35,50)(36,49)(37,51)(39,52)(42,53)(44,54)(46,55)(47,56);(1,3,4,8)(2,6,5,7)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(27,37)(28,40)(32,46)(34,49)(39,42)(41,43)(44,47)(48,50)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6710165);