#5939264

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=i,d^2=fgi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef

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, 5846460);