#8933107

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	#47881	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840303);