#95003

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₄³	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

576 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=hi,d^2=gh,e^2=i,f^2=g,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,e^b=eh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2199);