#8908393

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815589);