#249174

order	512
nilpotency class	3
isoclinism	isoclinic class of #249139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁶	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156370);