#6437450

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437445
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
prime radical	#62811	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6116 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,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	#72280406, #72302287
maximal normal subgroup of	#72280406, #72302287
maximal quotient of	#72302322

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344646);