#6880117

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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

1237 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=fh,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,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6787313);