#509572

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

541 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=dei,f^c=fh,h^c=hi,f^d=fh,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, 416768);