#462321

order	512
nilpotency class	3
isoclinism	isoclinic class of #458069
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2653
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16426
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369517);