#414425

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1788 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)(6,8)(18,22)(20,23)(24,29)(25,30)(26,32)(28,31);(9,10)(11,14)(12,13)(15,16)(17,18,23,30,21,25,20,22)(19,26,31,29,27,24,28,32);(1,2,3,4)(10,13)(14,16)(17,19,21,27)(18,24,25,26)(20,28,23,31)(22,29,30,32);(5,6,7,8)(9,11,12,15)(10,14,13,16)(19,27)(24,26)(28,31)(29,32)

Polycyclic presentation:

2^9|b^2=ei,c^2=fi,d^2=ghi,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, 321621);