#8930298

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	#57118	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

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, 8837494);