#115887

order	512
nilpotency class	3
isoclinism	isoclinic class of #100101
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,e^2=i,f^2=hi,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fhi,e^c=ei,f^c=fhi,g^c=ghi

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, 23083);