#6629397

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628892
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	#32899	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4159 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,22)(20,31)(21,32)(23,36)(24,29)(25,30)(26,34)(27,35)(28,37)(33,38)(39,40);(2,5)(6,8)(9,11,14,22)(10,15,18,19)(12,20,24,34)(13,21,25,35)(16,26,29,31)(17,27,30,32)(23,33,37,40)(28,38,36,39);(9,12)(10,16)(11,20)(13,23)(14,24)(15,26)(17,28)(18,29)(19,31)(21,33)(22,34)(25,37)(27,38)(30,36)(32,39)(35,40);(9,13)(10,17)(11,21)(12,23)(14,25)(15,27)(16,28)(18,30)(19,32)(20,33)(22,35)(24,37)(26,38)(29,36)(31,39)(34,40);(1,3,4,7)(2,6,5,8)(10,18)(15,19)(16,29)(17,30)(26,31)(27,32)(28,36)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311275, #72311281, #72311315, …
maximal normal subgroup of	#72311275, #72311281, #72311315, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536593);