#6463037

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443052
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60645	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370233);