#6667397

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660402
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1349 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=di,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6574593);