#7194163

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=fg,d^2=f,e^2=hi,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=dhi,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, 7101359);