#261374

order	512
nilpotency class	3
isoclinism	isoclinic class of #246079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1190 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)(5,6,9,7)(8,11,10,12)(13,14)(15,18)(16,19)(17,20)(21,26)(22,25)(23,27)(24,28);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,18)(16,21)(17,22)(19,26)(20,25)(23,28)(24,27);(6,11)(7,12)(8,10)(13,16,17,23)(14,19,20,27)(15,21,22,28)(18,24,25,26);(1,3,4,2)(5,8,9,10)(6,12,7,11)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 168570);