#188166

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 95362);