#7196213

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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

1363 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgh,e^2=h,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=dh,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, 7103409);