#6465684

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444408
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

3092 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)(5,6,9,15,11,16,10,13)(7,17,18,26,21,28,20,12)(8,14,24,32,23,31,25,29)(19,33,34,36,35,30,22,27);(2,4)(5,7)(6,12)(8,22)(9,20)(10,18)(11,21)(13,17)(14,30)(15,28)(16,26)(19,25)(23,34)(24,35)(27,29)(31,33)(32,36);(6,13,16,15)(7,18,21,20)(12,26)(14,29,31,32)(17,28)(19,34,35,22)(27,36)(30,33);(5,8,10,25,11,23,9,24)(6,14,13,29,16,31,15,32)(7,19,20,22,21,35,18,34)(12,27,28,30,26,36,17,33);(8,23)(14,31)(19,35)(22,34)(24,25)(27,36)(29,32)(30,33)

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=h,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^d=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, 6372880);