#269032

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

974 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,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, 176228);