#6465599

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443052
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3058 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6372795);