#6573683

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4572 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341471, #72341707, #72341788
maximal normal subgroup of	#72341471, #72341707, #72341788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480879);