#457560

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

894 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=ei,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,f^c=fi,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, 364756);