#6512675

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499814
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35328	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=i,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6419871);