#233171

order	512
nilpotency class	3
isoclinism	isoclinic class of #233152
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1278 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)(5,6,10,14)(7,15,18,19)(8,13,9,12)(11,17,20,16)(22,27)(23,30)(24,25)(29,34)(31,35)(32,33);(2,4)(5,7,10,18)(6,11,14,20)(8,17,9,16)(12,19,13,15)(21,22,26,29)(23,31,33,35)(24,34,25,27)(28,30,36,32);(7,16)(8,9)(11,15)(12,13)(17,18)(19,20)(21,23,26,33)(22,28,29,36)(24,30,25,32)(27,35,34,31);(22,29)(27,34)(28,36)(31,35)

Polycyclic presentation:

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

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, 140367);