#7325633

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	#45093	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,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=dh,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, 7232829);