#6437474

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437473
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

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

6102 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)(7,17)(8,21)(9,16)(10,15)(11,14)(12,19)(13,22)(18,26)(20,29)(23,30)(24,27)(25,28)(31,33)(32,35)(34,36);(1,2)(3,4)(5,7,11,20)(6,12,14,27)(8,18,23,34)(9,24,10,19)(13,26,28,36)(15,29,16,17)(21,35,30,31)(22,33,25,32);(2,4)(5,8)(6,13)(7,18)(9,22)(10,25)(11,23)(12,26)(14,28)(15,21)(16,30)(17,31)(19,32)(20,34)(24,33)(27,36)(29,35);(2,4);(6,14)(12,27)(13,28)(15,16)(17,29)(21,30)(26,36)(31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312539, #72312585, #72312596
maximal normal subgroup of	#72312539, #72312585, #72312596

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344670);