#249248

order	512
nilpotency class	3
isoclinism	isoclinic class of #249199
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
prime radical	#36957	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1388 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,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, 156444);