#7325689

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1740 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)(3,7)(4,6)(5,8)(17,18,23,27)(19,28,31,39)(20,25,34,44)(21,26,22,37)(24,35,40,30)(29,45,32,48)(33,42,36,43)(38,46,41,47);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16);(9,11)(10,13)(12,16)(14,15)(17,19,21,35,23,31,22,30)(18,24,37,39,27,40,26,28)(20,32,36,47,34,29,33,46)(25,41,42,45,44,38,43,48);(3,8)(5,7)(10,14)(11,16)(17,20)(18,25)(19,29)(21,36)(22,33)(23,34)(24,38)(26,43)(27,44)(28,45)(30,47)(31,32)(35,46)(37,42)(39,48)(40,41);(2,6)(5,7)(10,14)(13,15)(17,21,23,22)(18,26,27,37)(19,30,31,35)(20,33,34,36)(24,39,40,28)(25,42,44,43)(29,46,32,47)(38,45,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7232885);