#466686

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

892 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373882);