#7674643

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581839);