#7512324

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1508 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7419520);