#343412

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	C₂×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

833 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)(13,14,19,22,18,15,16,20)(17,25,24,23,27,28,26,21);(1,2)(3,4)(5,7,8,9)(6,10,12,11)(13,15,18,14)(16,20,19,22)(17,25,27,28)(21,26,23,24);(2,4)(6,11)(7,9)(10,12)(13,16,18,19)(14,20,15,22)(17,24,27,26)(21,25,23,28);(6,11)(7,9)(10,12)(13,17,18,27)(14,21,15,23)(16,24,19,26)(20,25,22,28)

Polycyclic presentation:

2^9|a^2=efhi,b^2=hi,c^2=h,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, 250608);