#188116

order	512
nilpotency class	3
isoclinism	isoclinic class of #188049
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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2416 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)(17,18,21,26,22,24,20,25)(19,27,28,32,29,23,30,31);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,31)(26,32);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(18,24)(20,21)(23,27)(28,30);(17,20,22,21)(18,25,24,26)(19,28,29,30)(23,31,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95312);