#233409

order	512
nilpotency class	3
isoclinism	isoclinic class of #233367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,g^b=ghi,e^c=eh,f^c=fi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140605);