#398133

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	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1704 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)(11,14)(12,13)(15,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,32)(28,30)(29,31);(10,13)(14,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,30)(26,32);(1,2,3,4)(18,24)(19,27)(20,22)(23,30)(28,29)(31,32);(6,8)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,25,24,26)(19,28,29,27)(23,30,31,32)

Polycyclic presentation:

2^9|c^2=fi,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=di,g^c=gi

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, 305329);