#244990

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1052 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)(7,15,8,12)(9,18,19,13)(11,17,20,16)(23,31)(24,25)(27,35)(29,30)(32,36)(33,34);(5,7,10,8)(6,11,14,20)(9,17,19,16)(12,18,15,13)(21,22)(23,32)(24,30)(25,29)(26,28)(27,33)(31,36)(34,35);(2,4)(5,8,10,7)(6,12,14,15)(9,17,19,16)(11,18,20,13)(21,23,26,34)(22,27,28,36)(24,33,25,31)(29,32,30,35);(6,13)(9,19)(11,12)(14,18)(15,20)(16,17)(22,28)(27,36)(29,30)(32,35)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152186);