#6465680

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444406
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

3146 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)(5,6,9,14,11,16,10,15)(7,17,20,12,19,28,21,27)(8,13,23,29,25,30,24,22)(18,31,34,26,33,36,35,32);(2,4)(5,6)(7,17)(8,22)(9,15)(10,14)(11,16)(12,21)(13,24)(18,32)(19,28)(20,27)(23,30)(25,29)(26,33)(31,35)(34,36);(5,7)(6,12,16,27)(8,18)(9,20)(10,21)(11,19)(13,26,30,32)(14,28,15,17)(22,31,29,36)(23,34)(24,35)(25,33);(5,8,9,23,11,25,10,24)(6,13,14,29,16,30,15,22)(7,18,20,34,19,33,21,35)(12,26,28,36,27,32,17,31);(7,19)(12,27)(17,28)(18,33)(20,21)(26,32)(31,36)(34,35)

Polycyclic presentation:

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

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, 6372876);