#8933196

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	#48493	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4199 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=h,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, 8840392);