#6342573

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	Aut(G) order 524288
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂²×2₋¹⁺⁴	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3841 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=hi,g^2=i,b^a=bf,c^a=ci,f^a=fh,g^b=gh,d^c=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249769);