#7622790

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
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

1222 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)(17,18,22,28,23,24,36,38,21,26,35,41,19,25,30,40)(20,31,33,45,34,39,46,48,32,27,42,43,29,37,44,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,23)(18,24,26,25)(20,29,32,34)(22,30,35,36)(27,37,31,39)(28,38,41,40)(33,44,42,46)(43,47,45,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,21,23)(18,25,26,24)(20,29,32,34)(22,30,35,36)(27,39,31,37)(28,40,41,38)(33,44,42,46)(43,48,45,47);(2,8)(6,9)(7,16)(14,15)(18,26)(24,25)(27,31)(28,41)(37,39)(38,40)(43,45)(47,48);(17,20,22,33,23,34,36,46,21,32,35,42,19,29,30,44)(18,27,28,43,24,37,38,47,26,31,41,45,25,39,40,48)

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529986);