#6442729

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437632
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	C₂⁴	C₂⁵
prime radical	#62842	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4836 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)(7,11)(8,12)(9,13)(10,14)(15,21)(16,22)(17,23)(18,25)(19,26)(20,28)(24,30)(27,31)(29,33)(32,35)(34,36);(5,7,10,17)(6,11,14,23)(8,15,19,30)(9,16,20,31)(12,24,26,21)(13,27,28,22)(18,29,32,36)(25,33,35,34);(1,2,3,4)(5,8)(6,12)(7,15)(9,18)(10,19)(11,21)(13,25)(14,26)(16,29)(17,30)(20,32)(22,33)(23,24)(27,34)(28,35)(31,36);(5,9)(6,13)(7,16)(8,18)(10,20)(11,22)(12,25)(14,28)(15,29)(17,31)(19,32)(21,33)(23,27)(24,34)(26,35)(30,36);(8,19)(12,26)(15,30)(18,32)(21,24)(25,35)(29,36)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267074, #72267083, #72267086, …
maximal normal subgroup of	#72267074, #72267083, #72267086, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349925);