#494034

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1048 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)(5,6)(7,10)(8,9)(11,12)(14,18)(15,21)(17,24)(19,27)(20,23)(22,28)(25,26);(6,9)(10,12)(13,14,16,18)(15,19,22,26)(17,20,23,24)(21,25,28,27);(13,15)(14,19)(16,22)(17,25)(18,26)(20,28)(21,24)(23,27);(5,7,8,11)(6,10,9,12)(15,22)(19,26)(21,28)(25,27)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=di,f^c=fh,h^c=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, 401230);