#7336132

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	#52970	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1456 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)(11,14)(12,15)(13,16)(17,18,22,24)(19,21,26,23)(20,25,29,31)(27,30,32,28);(1,2)(3,7)(4,6)(5,8)(9,11)(10,13)(12,14)(15,16)(18,24)(19,26)(25,31)(27,32);(1,3)(2,5)(4,8)(6,7)(9,12,15,10)(11,13,16,14)(17,19,23,18,22,26,21,24)(20,27,28,25,29,32,30,31);(2,6)(3,8)(9,13,15,14)(10,16,12,11)(17,20,22,29)(18,25,24,31)(19,27,26,32)(21,30,23,28);(2,6)(5,7)(9,14,15,13)(10,16,12,11)(17,21,22,23)(18,19,24,26)(20,28,29,30)(25,32,31,27)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=f,e^2=f,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, 7243328);