#261498

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	#39412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1243 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)(5,6,10,7)(8,12,9,11)(13,14,19,15)(16,24,28,25)(17,23,18,21)(20,27,22,26);(1,2,4,3)(5,7,10,6)(8,11,9,12)(13,15,19,14)(16,25,28,24)(17,23,18,21)(20,27,22,26);(6,11)(7,12)(8,9)(13,16,19,28)(14,20,15,22)(17,27,18,26)(21,25,23,24);(1,3,4,2)(5,8,10,9)(6,12,7,11)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=hi,d^2=ei,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, 168694);