#495358

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

986 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efgh,c^2=f,d^2=i,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,g^b=gh,h^b=hi

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, 402554);