#414564

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1266 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)(17,18,21,23)(19,25,26,32)(20,28,30,24)(22,29,31,27);(9,10,13,15)(11,16,12,14)(17,19,21,26)(18,22,23,31)(20,29,30,27)(24,32,28,25);(2,4)(9,11,13,12)(10,14,15,16)(19,26)(22,31)(25,32)(27,29);(5,6,7,8)(9,12,13,11)(10,14,15,16)(18,23)(22,31)(24,28)(25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=gh,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321760);