#383908

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1750 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^d=ei

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, 291104);