#7104216

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1380 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)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,38)(28,42)(29,35)(30,36)(31,37)(32,47)(33,41)(34,43)(39,53)(40,51)(44,49)(45,52)(46,50)(48,55)(54,56);(2,5)(3,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,41)(26,33,37,38)(28,39,29,44)(30,46,32,40)(34,48,52,54)(35,53,42,49)(36,51,47,50)(43,56,45,55);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28)(26,34)(27,39)(29,31)(30,43)(32,45)(33,48)(35,36)(37,52)(38,54)(40,55)(41,44)(42,47)(46,56)(49,50)(51,53);(1,3,4,7)(2,6,5,8)(12,20)(15,24)(17,19)(21,23)(25,29,31,28)(26,35,37,42)(27,39,41,44)(30,43,32,45)(33,49,38,53)(34,47,52,36)(40,56,46,55)(48,51,54,50);(10,16)(11,18)(15,24)(17,19)(25,30,31,32)(26,36,37,47)(27,40,41,46)(28,43,29,45)(33,50,38,51)(34,35,52,42)(39,55,44,56)(48,49,54,53)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,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=ehi,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7011412);