#8926495

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	#58805	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833691);