#6443354

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437999
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
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

5279 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280597, #72283442, #72283486, …
maximal normal subgroup of	#72280597, #72283442, #72283486, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350550);