#4262199

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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

1298 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:

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

Polycyclic presentation:

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

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, 4169395);