#233456

order	512
nilpotency class	3
isoclinism	isoclinic class of #233431
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

1308 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)(6,11)(7,15)(8,9)(13,14)(16,17)(18,19)(21,22,26,30)(23,31,34,35)(24,29,25,28)(27,32,36,33);(2,4)(5,6,10,14)(7,12,16,20)(8,11,9,13)(15,18,17,19)(23,32)(24,25)(27,35)(28,29)(31,36)(33,34);(5,7,10,16)(6,12,14,20)(8,15,9,17)(11,18,13,19)(21,23,26,34)(22,27,30,36)(24,32,25,33)(28,35,29,31);(5,8,10,9)(6,13,14,11)(7,15,16,17)(12,19,20,18)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=h,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,d^b=dh,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, 140652);