#188165

order	512
nilpotency class	3
isoclinism	isoclinic class of #188062
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂²×#562	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ 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)(3,6)(4,11,12,7)(5,8)(9,15,16,13)(10,14)(17,18,20,24,21,22,28,30)(19,25,27,32,26,23,29,31);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(18,22)(20,28)(23,25)(27,29);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,19,21,26)(18,23,22,25)(20,29,28,27)(24,32,30,31);(3,10)(6,14)(9,16)(13,15)(19,26)(23,25)(27,29)(31,32)

Polycyclic presentation:

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

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, 95361);