#6437932

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4675 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,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	#72312550, #72312585, #72312596
maximal normal subgroup of	#72312550, #72312585, #72312596

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345128);