#8978380

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	#47700	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8885576);