#6444298

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

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

2670 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,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, 6351494);