#7105536

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1407 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=fhi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7012732);