#343410

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
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	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

811 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=hi,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=defhi,c^b=cg,d^b=de,e^b=eh,f^c=fi,e^d=eh,f^d=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, 250606);