#424852

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1402 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ei,c^2=fi,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=dei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 332048);