#430933

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338129);