#8112464

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	#58269	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4029 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:

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

Polycyclic presentation:

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

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, 8019660);