#96221

order	512
nilpotency class	3
isoclinism	isoclinic class of #95192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

568 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gh,d^2=gi,e^2=gi,f^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fghi,c^b=ch,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, 3417);