#6465595

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443048
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3078 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=hi,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

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, 6372791);