#6430992

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2679	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gh,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dfi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72340587
maximal subgroup of	#72340560, #72340572, #72340587, …
maximal normal subgroup of	#72340560, #72340572, #72340587, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6338188);