#6442709

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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

4456 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:

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

Polycyclic presentation:

2^9|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^b=fi,h^b=hi,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, 6349905);