#6442606

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437473
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
prime radical	#62812	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4579 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,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	#72312715, #72313632, #72313645
maximal normal subgroup of	#72312715, #72313632, #72313645

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349802);