#8192504

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#58177	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8099700);