#6443313

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

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×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5064 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,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	#72286980, #72286987, #72286994, …
maximal normal subgroup of	#72286980, #72286987, #72286994, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350509);