#6443351

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437995
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	#62812	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

5038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284251, #72284505, #72286176
maximal normal subgroup of	#72284251, #72284505, #72286176

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350547);