#249306

order	512
nilpotency class	3
isoclinism	isoclinic class of #249259
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	#2617	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1192 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156502);