#498388

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

550 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)(13,14,18,22,19,15,23,25,17,21,26,28,16,24,20,27);(5,6)(7,11)(8,12)(9,10)(13,15,18,25,19,21,23,28,17,24,26,27,16,14,20,22);(2,4)(5,7,9,8)(6,11,10,12)(13,16,17,19)(14,15,21,24)(18,20,26,23)(22,25,28,27);(5,8,9,7)(6,11,10,12)(13,16,17,19)(14,15,21,24)(18,20,26,23)(22,25,28,27)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405584);