#8931033

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	#46076	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3833 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)(7,14)(10,12)(15,16)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(34,39)(35,37)(41,44)(43,45);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(21,29)(24,28)(26,32)(30,31)(33,34,38,39)(35,40,37,42)(36,44,46,41)(43,48,45,47);(2,6)(3,9)(8,11)(10,12)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,3)(2,7)(4,12)(5,9)(6,14)(8,16)(10,13)(11,15)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,35,38,37)(34,40,39,42)(36,43,46,45)(41,47,44,48);(1,4,5,13)(2,8,6,11)(3,10,9,12)(7,15,14,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,37,38,35)(34,42,39,40)(36,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=h,e^2=gi,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, 8838229);