#6992527

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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

1177 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6899723);