#8757133

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	#55585	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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:

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

Polycyclic presentation:

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

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, 8664329);