#7203715

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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

1398 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)(3,5)(4,6)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,31,35)(27,36,37,48)(28,40,41,33)(29,34,30,44)(32,45,46,39)(38,53,52,51)(42,50,43,49)(47,55,56,54);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(27,37)(28,41)(32,46)(33,40)(36,48)(39,45)(42,43)(49,50);(3,7)(5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,31,37)(26,32,35,46)(28,38,41,52)(29,39,30,45)(33,47,40,56)(34,48,44,36)(42,54,43,55)(49,51,50,53);(2,6)(3,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,38)(29,43)(30,42)(31,41)(32,47)(34,50)(35,40)(36,51)(37,52)(39,55)(44,49)(45,54)(46,56)(48,53);(3,7)(5,8)(10,18)(15,19)(16,17)(23,24)(25,29,31,30)(26,34,35,44)(27,39,37,45)(28,42,41,43)(32,48,46,36)(33,49,40,50)(38,54,52,55)(47,51,56,53)

Polycyclic presentation:

2^9|a^2=i,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,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, 7110911);