#7510727

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

1267 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)(19,28)(20,33)(21,25)(22,26)(23,27)(24,31)(29,39)(30,42)(32,45)(34,46)(35,40)(36,43)(37,44)(38,41)(47,48);(1,3)(2,6)(4,7)(5,8)(17,19,22,31)(18,24,26,28)(20,30,21,35)(23,38,37,32)(25,42,33,40)(27,45,44,41)(29,36,48,34)(39,46,47,43);(10,13)(14,15)(17,20,22,21)(18,25,26,33)(19,29,31,48)(23,36,37,34)(24,39,28,47)(27,46,44,43)(30,32,35,38)(40,45,42,41);(2,7)(3,8)(9,11,12,16)(10,14,13,15)(17,21,22,20)(18,25,26,33)(19,30,31,35)(23,36,37,34)(24,40,28,42)(27,43,44,46)(29,32,48,38)(39,45,47,41);(1,4,5,6)(2,3,7,8)(20,21)(25,33)(29,48)(30,35)(34,36)(39,47)(40,42)(43,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=g,e^2=fi,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, 7417923);