#384203

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

1162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fi,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, 291399);