#8095782

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	#45514	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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gi,e^2=ghi,f^2=h,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, 8002978);