#269036

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1436 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:

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

Polycyclic presentation:

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

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, 176232);