#475885

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=e,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=di,d^c=de,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383081);