#423034

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
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×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1478 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330230);