#3517771

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1368 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fg,d^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,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, 3424967);