#367573

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 20 points:

(2,3)(10,12)(13,14,18,15,17,19,16,20);(1,2,4,3)(5,6,7,8)(10,12)(13,15,18,19,17,20,16,14);(1,2,4,3)(6,8)(9,10,11,12);(1,3,4,2)(13,16,17,18)(14,15,19,20)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=f,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274769);