#8930178

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	#51952	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3915 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)(17,18,22,23)(19,20,24,27)(21,25,30,31)(26,28,29,32)(33,34)(35,40)(36,45)(37,38)(39,42)(41,48)(43,47)(44,46);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(18,23)(20,27)(25,31)(28,32)(33,35,39,36)(34,40,42,45)(37,46,41,43)(38,44,48,47);(2,6)(3,9)(8,11)(10,12)(17,19,22,24)(18,20,23,27)(21,29,30,26)(25,32,31,28)(33,36,39,35)(34,40,42,45)(37,46,41,43)(38,47,48,44);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,19)(18,20)(21,26)(22,24)(23,27)(25,28)(29,30)(31,32);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,20)(18,24)(19,23)(21,28)(22,27)(25,29)(26,31)(30,32)(33,37,39,41)(34,38,42,48)(35,43,36,46)(40,47,45,44);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,21,22,30)(18,25,23,31)(19,26,24,29)(20,28,27,32)(33,38)(34,41)(35,44)(36,47)(37,42)(39,48)(40,43)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=i,e^2=h,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, 8837374);