#210034

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117230);