#6443285

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dhi,f^c=fi,h^c=hi,f^d=fi,h^d=hi

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, 6350481);