#6437795

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437793
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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 ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4597 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)(3,4)(5,6,10,17,12,16,11,18)(7,19,22,30,24,31,23,13)(8,14,25,33,26,32,9,15)(20,29,21,34,27,36,35,28);(5,7)(6,13)(8,21)(9,27)(10,23)(11,22)(12,24)(14,29)(15,34)(16,30)(17,31)(18,19)(20,25)(26,35)(28,33)(32,36);(2,4)(5,8,12,26)(6,14,16,32)(7,20,24,27)(9,10,25,11)(13,28,30,34)(15,17,33,18)(19,29,31,36)(21,23,35,22);(5,9)(6,15)(7,21)(8,10)(11,26)(12,25)(13,29)(14,17)(16,33)(18,32)(19,34)(20,23)(22,27)(24,35)(28,31)(30,36);(6,16)(13,30)(14,32)(15,33)(17,18)(19,31)(28,34)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238736, #72264906, #72264949, …
maximal normal subgroup of	#72238736, #72264906, #72264949, …
maximal quotient of	#72238755

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344991);