#249194

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	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1243 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^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, 156390);