#261462

order	512
nilpotency class	3
isoclinism	isoclinic class of #246239
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	C₂×#534	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1142 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)(7,8)(9,11)(10,12)(13,14)(15,20)(16,25)(17,22)(18,23)(19,24)(21,27)(26,28);(1,2,4,3)(5,7,10,11)(6,8,12,9)(13,15,19,25)(14,20,24,16)(17,21,18,28)(22,27,23,26);(6,7)(8,9)(11,12)(13,16,19,20)(14,21,24,28)(15,22,25,23)(17,26,18,27);(1,3,4,2)(5,8,10,9)(6,7,12,11)(13,17,19,18)(14,22,24,23)(15,21,25,28)(16,26,20,27)

Polycyclic presentation:

2^9|b^2=eh,c^2=hi,d^2=eh,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=di,f^c=fh,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 168658);