#6437968

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437963
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	#62811	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4686 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316311, #72316331, #72316343
maximal normal subgroup of	#72316311, #72316331, #72316343

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345164);