#465240

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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

734 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egi,c^2=f,d^2=e,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, 372436);