#188154

order	512
nilpotency class	3
isoclinism	isoclinic class of #188035
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2360 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,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, 95350);