#373983

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
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

1018 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=hi,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,d^c=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, 281179);