#7332352

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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

1339 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7239548);