#7300262

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234421
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1205 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34,38,42)(35,39,44,48)(36,41,37,40)(43,47,45,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(19,25)(20,27)(22,28)(23,30)(33,35)(34,39)(36,43)(37,45)(38,44)(40,46)(41,47)(42,48);(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(34,40)(36,37)(39,46)(41,42)(43,45)(47,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,25)(22,28)(26,31)(29,32)(33,36,38,37)(34,41,42,40)(35,43,44,45)(39,47,48,46);(2,8)(6,9)(7,16)(14,15)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(35,44)(39,48)(43,45)(46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi

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, 7207458);