#7338701

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324820
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	#24256	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(9,10,14,11)(12,15,13,16)(17,18)(19,27)(20,28)(21,24)(22,25)(23,30)(26,29)(31,32);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,15)(13,16)(17,19,22,28)(18,20,25,27)(21,29,31,23)(24,30,32,26);(1,3)(2,5)(4,8)(6,7)(9,11,14,10)(12,15,13,16)(17,20,29,32,22,27,23,24)(18,21,30,19,25,31,26,28);(2,6)(3,8)(9,12,14,13)(10,15,11,16)(17,21,22,31)(18,24,25,32)(19,23,28,29)(20,26,27,30);(2,6)(5,7)(9,13,14,12)(10,15,11,16)(18,25)(19,28)(21,31)(26,30)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,e^2=fi,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=dhi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7245897);