#8416829

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3775 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)(7,10)(11,13)(15,16)(17,18)(19,25)(20,26)(21,24)(22,23)(27,31)(28,32)(29,30)(33,34,38,42)(35,39,45,48)(36,37,41,40)(43,44,47,46);(1,2)(3,10)(4,12)(5,6)(7,9)(8,14)(11,15)(13,16)(18,23)(19,26)(24,30)(27,32)(33,35,38,45)(34,39,42,48)(36,43,41,47)(37,44,40,46);(2,6)(7,10)(8,12)(15,16)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,36,38,41)(34,40,42,37)(35,43,45,47)(39,46,48,44);(1,3,5,9)(2,7,6,10)(4,13,14,11)(8,16,12,15)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,35)(34,39)(36,43)(37,44)(38,45)(40,46)(41,47)(42,48);(1,4,5,14)(2,8,6,12)(3,11,9,13)(7,15,10,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,21,22,29)(18,24,23,30)(19,27,26,32)(20,28,25,31)(33,37,38,40)(34,41,42,36)(35,44,45,46)(39,47,48,43)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8324025);