#268516

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
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	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1129 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,d^2=h,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,f^c=fhi,e^d=ei,f^d=fhi

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, 175712);