#8930495

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	#58248	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3979 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)(19,21)(20,22)(23,25)(24,28)(26,30)(27,31)(29,32)(33,34,39,37)(35,36,42,40)(38,45,46,41)(43,47,48,44);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,19,23,28)(18,24,25,21)(20,26,29,31)(22,27,32,30)(34,37)(35,42)(41,45)(43,48);(2,6)(3,9)(8,11)(10,12)(20,29)(22,32)(26,31)(27,30)(33,35,39,42)(34,40,37,36)(38,43,46,48)(41,47,45,44);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,20)(18,22)(19,26)(21,30)(23,29)(24,27)(25,32)(28,31)(33,36)(34,35)(37,42)(38,44)(39,40)(41,43)(45,48)(46,47);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,21,23,24)(18,19,25,28)(20,27,29,30)(22,31,32,26)(33,37,39,34)(35,36,42,40)(38,45,46,41)(43,44,48,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,22)(18,20)(19,27)(21,31)(23,32)(24,26)(25,29)(28,30)(33,38,39,46)(34,41,37,45)(35,43,42,48)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=hi,e^2=g,f^2=h,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, 8837691);