#8875592

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	#54394	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3970 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8782788);