#408004

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	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1602 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=di,e^c=ei,f^c=fi,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, 315200);