#370409

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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

1046 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=fgi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di

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, 277605);