#8095147

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	#52963	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 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)(3,9)(4,11)(5,13)(6,8)(7,14)(10,12)(15,16)(20,28)(24,29)(26,32)(30,31)(33,34)(35,40)(36,42)(37,41)(38,43)(39,46)(44,47)(45,48);(1,3,6,11)(2,4,8,9)(5,10,14,16)(7,12,13,15)(18,22)(23,25)(24,29)(30,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,43)(41,47)(42,48);(1,2,6,8)(3,9,11,4)(5,7,14,13)(10,15,16,12)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,35)(34,40)(36,44)(37,45)(38,46)(39,43)(41,48)(42,47);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(33,36,38,37)(34,41,43,42)(35,44,46,45)(39,47,40,48);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,14)(15,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,34)(35,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5,6,14)(2,7,8,13)(3,10,11,16)(4,12,9,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,36)(34,42,43,41)(35,45,46,44)(39,48,40,47)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002343);