#398512

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1348 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)(7,8)(9,10,13,15)(11,16,12,14)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,31)(29,32);(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,21)(22,28)(23,29)(25,30)(26,31)(27,32);(1,2,3,4)(17,20,21,19,23,28,22,29)(18,24,26,30,27,32,25,31);(6,8)(9,12,13,11)(10,14,15,16)(17,21,23,22)(18,25,27,26)(19,20,29,28)(24,30,32,31)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=f,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,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, 305708);