#249072

order	512
nilpotency class	3
isoclinism	isoclinic class of #249019
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

1263 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)(7,9,12,17)(8,11,16,19)(14,20,18,15)(21,22)(23,28)(24,33)(25,29)(26,34)(27,30)(31,35)(32,36);(1,2)(3,4)(5,7)(6,9)(8,14)(10,12)(11,15)(13,17)(16,18)(19,20)(21,23,27,32)(22,28,30,36)(24,26,31,25)(29,33,34,35);(2,4)(5,8)(6,11)(7,14)(9,15)(10,16)(12,18)(13,19)(17,20)(21,24)(22,28)(23,25)(26,32)(27,31)(29,35)(30,36)(33,34);(5,9)(6,12)(7,13)(8,15)(10,17)(11,18)(14,19)(16,20)(21,25,27,26)(22,29,30,34)(23,31,32,24)(28,35,36,33)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156268);