#6463105

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,23)(16,21)(17,29)(18,20)(19,25)(22,30)(24,36)(26,35)(27,33)(28,41)(31,39)(32,38)(34,40)(37,43)(42,44);(7,8)(10,12)(13,15,17,25,18,26,30,39)(14,19,22,23,20,31,29,35)(16,24,28,38,27,37,40,44)(21,32,34,36,33,42,41,43);(1,2,3,4)(5,7,9,8)(6,10,11,12)(14,20)(19,31)(21,33)(22,29)(23,35)(32,42)(34,41)(36,43);(5,8,9,7)(6,10,11,12)(13,16)(14,21)(15,24)(17,28)(18,27)(19,32)(20,33)(22,34)(23,36)(25,38)(26,37)(29,41)(30,40)(31,42)(35,43)(39,44);(16,27)(21,33)(24,37)(28,40)(32,42)(34,41)(36,43)(38,44)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370301);