#268395

order	512
nilpotency class	3
isoclinism	isoclinic class of #268355
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

998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,12,20,10,15,11,14)(7,13,18,17,9,8,16,19)(21,22,25,29)(23,30,32,36)(24,26,33,28)(27,34,35,31);(1,2)(3,4)(5,7,10,9)(6,13,15,8)(11,18,12,16)(14,17,20,19)(22,26)(24,33)(27,30)(28,29)(31,34)(35,36);(2,4)(5,8)(6,7)(9,15)(10,13)(11,19)(12,17)(14,18)(16,20);(5,9,10,7)(6,8,15,13)(11,16,12,18)(14,19,20,17)(21,23,25,32)(22,27,29,35)(24,34,33,31)(26,30,28,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,d^2=h,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=de,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175591);