#8093575

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3781 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)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34,38,39)(35,43,46,40)(36,37,42,41)(44,47,48,45);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(18,22)(23,25)(24,29)(30,31)(34,39)(36,42)(40,43)(44,48);(2,6)(7,9)(8,13)(14,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,38,46)(34,40,39,43)(36,44,42,48)(37,45,41,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,42)(34,41,39,37)(35,44,46,48)(40,47,43,45);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37,38,41)(34,42,39,36)(35,45,46,47)(40,48,43,44)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=g,e^2=ghi,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, 8000771);