#398579

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=efi,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, 305775);