#7105518

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

1410 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)(27,32)(28,38)(29,34)(30,43)(31,35)(33,36)(37,46)(39,52)(40,45)(41,49)(42,50)(44,48)(47,51)(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)(25,27,31,30)(26,32,35,43)(28,36,40,44)(29,37,42,41)(33,45,48,38)(34,46,50,49)(39,51,54,53)(47,55,56,52);(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)(27,36)(29,39)(30,44)(31,40)(32,45)(34,47)(35,48)(37,51)(38,43)(41,53)(42,54)(46,55)(49,52)(50,56);(1,4,5,2)(3,6,8,7)(12,20)(15,24)(17,19)(21,23)(25,29,31,42)(26,34,35,50)(27,37,30,41)(28,39,40,54)(32,46,43,49)(33,47,48,56)(36,51,44,53)(38,52,45,55);(10,16)(11,18)(15,24)(17,19)(25,27,31,30)(26,32,35,43)(28,36,40,44)(29,41,42,37)(33,45,48,38)(34,49,50,46)(39,53,54,51)(47,52,56,55)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=fh,d^2=fi,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, 7012714);