#95352

order	512
nilpotency class	3
isoclinism	isoclinic class of #95192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=gi,e^2=hi,f^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fghi,d^b=dh,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2548);