#5236739

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
automorphism	Aut(G) order 2097152
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1598 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)(5,8)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(24,29)(27,31)(33,34,38,35)(36,41,43,39)(37,40,45,42)(44,47,48,46);(1,2,4,6)(3,5,7,8)(11,15)(13,16)(20,28)(24,29)(27,31)(30,32)(34,35)(36,43)(40,42)(44,48);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,38,34)(36,39,43,41)(37,40,45,42)(44,47,48,46);(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=f,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5143935);