#7330703

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	#30851	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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)(17,18,24,21)(19,22,26,23)(20,28,30,27)(25,32,31,29);(1,2)(3,7)(4,6)(5,8)(9,11,13,10)(12,15,16,14)(17,19,24,26)(18,22,21,23)(20,28,30,27)(25,29,31,32);(1,3)(2,5)(4,8)(6,7)(10,11)(12,16)(17,20,23,29,24,30,22,32)(18,25,26,28,21,31,19,27);(2,6)(3,8)(9,12,13,16)(10,14,11,15)(17,21,24,18)(19,22,26,23)(20,25,30,31)(27,29,28,32);(2,6)(5,7)(10,11)(14,15)(17,22,24,23)(18,26,21,19)(20,29,30,32)(25,27,31,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fg,d^2=fi,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, 7237899);