#7167432

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1369 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=dfi,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, 7074628);