#2774121

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

(3,7)(5,8)(9,10,12,14)(11,13,15,16)(17,18,22,19)(20,23,27,25)(21,26,30,24)(28,32,29,31)(34,35)(39,41)(40,42)(47,48);(11,15)(13,16)(18,19)(20,27)(24,26)(28,29)(33,34,38,35)(36,39,44,41)(37,40,46,42)(43,47,45,48);(1,2,4,6)(3,5,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,18)(20,23,27,25)(21,26,30,24)(28,31,29,32)(33,35,38,34)(36,39,44,41)(37,42,46,40)(43,47,45,48);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,27)(18,23,19,25)(21,29,30,28)(24,32,26,31)(33,36)(34,39)(35,41)(37,45)(38,44)(40,48)(42,47)(43,46);(17,21,22,30)(18,24,19,26)(20,28,27,29)(23,31,25,32)(33,37,38,46)(34,40,35,42)(36,43,44,45)(39,47,41,48)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2681317);