#5118140

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1425 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)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,34,37,40)(35,41,42,38)(36,44,45,39)(43,46,48,47);(2,5)(6,8)(9,11,14,13)(10,15,12,16)(18,19)(20,27)(24,26)(28,32)(35,42)(36,45)(38,41)(39,44);(1,2,4,5)(3,6,7,8)(9,12,14,10)(11,15,13,16)(17,19,22,18)(20,23,27,25)(21,24,29,26)(28,31,32,30)(33,35)(34,38)(36,43)(37,42)(39,46)(40,41)(44,47)(45,48);(9,11,14,13)(10,16,12,15)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,34)(35,41)(36,39)(37,40)(38,42)(43,47)(44,45)(46,48);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,15,12,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=gh,d^2=gi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=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, 5025336);