#5178020

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1360 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fh,d^2=g,e^2=fghi,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, 5085216);