#8932112

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	#51768	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3858 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839308);