#403200

order	512
nilpotency class	3
isoclinism	isoclinic class of #394183
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1116 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=f,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,d^c=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, 310396);