#405078

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1164 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)(9,10,13,12)(11,14,15,16)(18,23)(20,21)(24,31)(25,26)(27,32)(28,30);(9,11)(10,14)(12,16)(13,15)(17,18,20,25,22,26,21,23)(19,27,30,31,29,24,28,32);(1,2,3,4)(11,15)(14,16)(17,19,22,29)(18,24,26,27)(20,30,21,28)(23,31,25,32);(6,8)(9,12,13,10)(11,16,15,14)(17,20,22,21)(18,25,26,23)(19,28,29,30)(24,31,27,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=ei,c^2=fi,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,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, 312274);