#188047

order	512
nilpotency class	3
isoclinism	isoclinic class of #188035
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#553	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,g^b=gh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95243);