#8930847

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	#54935	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,e^2=i,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, 8838043);