#409072

order	512
nilpotency class	3
isoclinism	isoclinic class of #394177
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

1388 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fi,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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316268);