#391432

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

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

888 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298628);