#498427

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂×#121	C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

658 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)(6,10)(11,12)(13,14,17,23,18,20,25,27,16,22,19,26,15,21,24,28);(5,6,9,10)(7,11,8,12)(13,14,19,26,18,20,24,28,16,22,17,23,15,21,25,27);(2,4)(5,7,9,8)(6,11,10,12)(13,15,16,18)(14,20,22,21)(17,24,19,25)(23,27,26,28);(5,8,9,7)(6,11,10,12)(13,15,16,18)(14,21,22,20)(17,24,19,25)(23,28,26,27)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=ci,d^b=dei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405623);