#8093573

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	#53106	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(17,18,22,23)(19,26,25,21)(20,24,30,31)(27,32,28,29)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,47)(44,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19)(18,21)(20,28)(22,25)(23,26)(24,32)(27,30)(29,31)(34,39)(36,40)(41,45)(44,47);(2,6)(7,9)(8,13)(14,15)(18,23)(19,25)(24,31)(27,28)(33,35,38,43)(34,36,39,40)(37,42,46,48)(41,44,45,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,18)(19,26)(20,24)(21,25)(22,23)(27,32)(28,29)(30,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,20,22,30)(18,24,23,31)(19,27,25,28)(21,29,26,32)(33,36,38,40)(34,35,39,43)(37,44,46,47)(41,42,45,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,26)(18,25,23,19)(20,29,30,32)(24,28,31,27)(33,37,38,46)(34,41,39,45)(35,42,43,48)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,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, 8000769);