#8093571

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	#53101	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=g,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^b=eghi,d^c=dghi

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, 8000767);