#7409485

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1234 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)(3,6,7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26)(27,36)(28,39)(29,34)(30,35)(31,32)(33,37)(38,50)(40,49)(41,55)(42,47)(43,46)(44,56)(45,52)(48,53)(51,54);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,18)(22,23)(25,27,30,31)(26,32,35,36)(28,40,42,37)(29,43,45,38)(33,47,49,39)(34,50,52,46)(41,53,44,54)(48,55,51,56);(2,4)(3,7)(25,28,30,42)(26,33,35,49)(27,37,31,40)(29,44,45,41)(32,39,36,47)(34,51,52,48)(38,54,43,53)(46,56,50,55);(1,4,5,2)(3,6,7,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,29)(26,34)(27,38)(28,41)(30,45)(31,43)(32,46)(33,48)(35,52)(36,50)(37,53)(39,55)(40,54)(42,44)(47,56)(49,51);(3,7)(6,8)(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(26,35)(29,45)(32,36)(33,49)(38,43)(39,47)(41,44)(53,54)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7316681);