#495982

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

606 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,d^2=eg,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, 403178);