#7330733

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324532
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	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1504 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)(11,13)(12,14)(15,16)(17,18,21,23)(19,27,28,32)(20,25,22,26)(24,30,31,29);(1,2)(3,7)(4,6)(5,8)(9,11,13,10)(12,15,16,14)(18,23)(19,28)(25,26)(29,30);(1,3)(2,5)(4,8)(6,7)(10,11)(12,16)(17,19,20,30,21,28,22,29)(18,24,26,32,23,31,25,27);(2,6)(3,8)(9,12,13,16)(10,14,11,15)(17,20,21,22)(18,25,23,26)(19,29,28,30)(24,32,31,27);(2,6)(5,7)(10,11)(14,15)(17,20,21,22)(18,26,23,25)(19,30,28,29)(24,32,31,27)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7237929);