#7510728

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1271 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)(3,4)(5,7)(6,8)(9,10,12,13)(11,15,16,14)(17,18,23,28)(19,24,32,41)(20,25,21,26)(22,27,37,46)(29,38,47,48)(30,39,33,42)(31,40,36,45)(34,43,35,44);(1,3)(2,6)(4,7)(5,8)(17,19,23,32)(18,24,28,41)(20,33,21,30)(22,36,37,31)(25,42,26,39)(27,45,46,40)(29,35,47,34)(38,44,48,43);(10,13)(14,15)(17,20,23,21)(18,25,28,26)(19,29,32,47)(22,35,37,34)(24,38,41,48)(27,44,46,43)(30,36,33,31)(39,45,42,40);(2,7)(3,8)(9,11,12,16)(10,14,13,15)(17,21,23,20)(18,26,28,25)(19,30,32,33)(22,35,37,34)(24,39,41,42)(27,44,46,43)(29,36,47,31)(38,45,48,40);(1,4,5,6)(2,3,7,8)(20,21)(25,26)(29,47)(30,33)(34,35)(38,48)(39,42)(43,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7417924);