#6372977

order	512
nilpotency class	3
isoclinism	isoclinic class of #60470
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2688 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280173);