#509531

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

531 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=ei,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=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, 416727);