#504660

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13304	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

615 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411856);