#367574

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

1154 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:

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

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,d^c=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, 274770);