#8926576

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3776 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)(3,12,4,8)(5,10,11,6)(13,14,15,16)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(34,40)(35,45)(36,38)(43,48);(1,3,7,4)(2,8,9,12)(5,13,11,15)(6,14,10,16)(20,28)(23,30)(26,32)(29,31)(33,34)(35,46)(36,42)(37,48)(38,44)(39,40)(41,45)(43,47);(2,9)(6,10)(8,12)(14,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,35)(34,41)(36,37)(38,47)(39,45)(40,46)(42,43)(44,48);(1,4,7,3)(2,8,9,12)(5,13,11,15)(6,16,10,14)(33,36,39,38)(34,42,40,44)(35,37,45,47)(41,43,46,48);(1,5,7,11)(2,10,9,6)(3,13,4,15)(8,16,12,14)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,37,39,47)(34,43,40,48)(35,36,45,38)(41,42,46,44);(1,6,7,10)(2,11,9,5)(3,14,4,16)(8,15,12,13)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,38,39,36)(34,44,40,42)(35,47,45,37)(41,48,46,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=g,e^2=h,f^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, 8833772);