#8202364

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	#37940	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4243 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)(3,9)(4,12)(5,7)(6,11)(8,14)(10,15)(13,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,34,38,41)(35,37,43,39)(36,40,45,47)(42,44,48,46);(1,3,5,11)(2,6,7,9)(4,13,14,10)(8,16,12,15)(19,20)(21,30)(23,26)(24,29)(33,35,38,43)(34,39,41,37)(36,42,45,48)(40,46,47,44);(2,7)(6,9)(8,12)(15,16)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,36,38,45)(34,40,41,47)(35,42,43,48)(37,44,39,46);(1,4,5,14)(2,8,7,12)(3,10,11,13)(6,15,9,16)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,37)(34,35)(36,44)(38,39)(40,42)(41,43)(45,46)(47,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,15)(12,16)(13,14)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,35)(34,37)(36,42)(38,43)(39,41)(40,44)(45,48)(46,47);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

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

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, 8109560);