#7326401

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324564
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	#53064	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1504 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)(17,18,22,27)(19,23,30,26)(20,31,34,38)(21,25,37,44)(24,32,41,28)(29,39,35,42)(33,47,36,48)(40,46,43,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,27,26)(20,32,34,28)(21,35,37,29)(24,31,41,38)(25,42,44,39)(33,45,36,46)(40,48,43,47);(2,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,30,32,22,34,19,28)(18,24,23,38,27,41,26,31)(21,36,35,45,37,33,29,46)(25,43,39,47,44,40,42,48);(3,8)(5,7)(10,14)(11,16)(17,21)(18,25)(19,29)(20,33)(22,37)(23,39)(24,40)(26,42)(27,44)(28,45)(30,35)(31,47)(32,46)(34,36)(38,48)(41,43);(2,6)(5,7)(10,14)(13,15)(17,19,22,30)(18,23,27,26)(20,28,34,32)(21,35,37,29)(24,38,41,31)(25,42,44,39)(33,46,36,45)(40,47,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,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=ehi,d^c=dh,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, 7233597);