#1907184

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

1388 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,d^2=fgh,e^2=fg,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, 1814380);