#6963348

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1241 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6870544);