#8295579

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	#53838	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3908 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8202775);