#8926715

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=gh,e^2=h,f^2=g,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, 8833911);