#8926557

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	#55823	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3842 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833753);