#249241

order	512
nilpotency class	3
isoclinism	isoclinic class of #249199
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

1203 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)(7,12)(8,17)(9,13)(10,18)(11,14)(15,19)(16,20)(21,22,26,25)(23,27,31,28)(24,32,35,30)(29,36,33,34);(1,2)(3,4)(5,7)(6,12)(8,9)(10,15)(11,16)(13,17)(14,20)(18,19)(21,23,26,31)(22,28,25,27)(24,30,35,32)(29,34,33,36);(2,4)(5,8)(6,12)(7,9)(10,16)(11,15)(13,19)(14,20)(17,18)(21,24)(22,29)(23,30)(25,33)(26,35)(27,36)(28,34)(31,32);(5,9,11,10)(6,13,14,18)(7,15,16,8)(12,19,20,17)(21,25,26,22)(23,28,31,27)(24,33,35,29)(30,34,32,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,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, 156437);