#6437998

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437995
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	#62811	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4778 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)(7,15)(8,19,21,12)(9,22)(10,14)(11,18)(13,23)(16,27,30,24)(17,25)(20,26,33,32)(28,31)(29,34,36,35);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,24)(13,25)(14,15)(19,27)(20,29)(21,30)(22,28)(23,31)(26,34)(32,35)(33,36);(2,4)(5,8)(6,12)(7,16)(9,20)(10,21)(11,24)(13,26)(14,19)(15,27)(17,29)(18,30)(22,32)(23,33)(25,34)(28,35)(31,36);(2,4);(5,9,10,23)(6,13,14,22)(7,17,18,31)(8,20,21,33)(11,25,15,28)(12,26,19,32)(16,29,30,36)(24,34,27,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281147, #72283147, #72283410
maximal normal subgroup of	#72281147, #72283147, #72283410

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345194);