#7622791

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁵
commutator, socle	C₂²	C₂⁴×C₈
prime radical	#3482	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1199 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)(17,18,21,26,22,20,32,33,23,27,35,41,36,34,25,40)(19,24,30,39,31,29,45,46,28,37,43,47,44,42,38,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,24)(20,29)(21,30)(22,31)(23,28)(25,38)(26,39)(27,37)(32,45)(33,46)(34,42)(35,43)(36,44)(40,48)(41,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,28)(24,37)(29,42)(30,43)(31,44)(38,45)(39,47)(46,48);(2,8)(6,9)(7,16)(14,15)(19,28)(24,37)(29,42)(30,43)(31,44)(38,45)(39,47)(46,48);(17,20,21,33,22,27,32,41,23,34,35,40,36,18,25,26)(19,29,30,46,31,37,45,47,28,42,43,48,44,24,38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,e^2=g,g^2=h,h^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cfi,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529987);