#6437974

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
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 ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4564 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312662, #72313645, #72313659
maximal normal subgroup of	#72312662, #72313645, #72313659

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345170);