#6438864

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60369	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3099 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^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=dh,e^b=ei,h^b=hi,g^c=gi,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, 6346060);