#1917162

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
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

1466 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,c^2=fghi,d^2=gh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 1824358);