#414174

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

1152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=e,d^2=gh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=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, 321370);