#8930399

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	#56782	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3903 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=i,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, 8837595);