#7489330

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1364 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26)(27,31)(28,33)(29,32)(30,35)(34,41)(36,44)(37,43)(38,45)(39,48)(40,46)(42,50)(47,52)(49,55)(51,54)(53,56);(2,6)(3,4)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,30,36)(26,31,35,44)(28,37,39,49)(29,38,41,50)(32,45,34,42)(33,43,48,55)(40,51,52,56)(46,54,47,53);(1,2)(3,8)(4,7)(5,6)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(27,36)(28,39)(29,41)(31,42)(33,47)(34,35)(43,54)(44,45)(46,48)(51,56)(53,55);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,37)(29,40)(30,39)(31,43)(32,46)(34,47)(35,48)(36,49)(38,51)(41,52)(42,53)(44,55)(45,54)(50,56);(1,4,5,3)(2,7,6,8)(10,18)(15,19)(16,17)(23,24)(27,36)(28,39)(31,44)(33,48)(38,50)(40,52)(42,45)(46,47)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,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, 7396526);