#7622754

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	#3475	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1245 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 7529950);