#460602

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1019 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367798);