#8926528

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	#46962	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3838 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=g,e^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, 8833724);