#188045

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2456 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)(3,11,13,7)(4,8)(5,9)(6,10)(12,16,14,15)(17,18)(19,22)(20,27,28,23)(21,24)(25,31,32,29)(26,30);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=h,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, 95241);