#6494412

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2498 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,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, 6401608);