#406442

order	512
nilpotency class	3
isoclinism	isoclinic class of #395154
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=fi,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313638);