#8184899

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	#34962	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4099 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,14)(15,16)(17,18,24,23)(19,21,25,20)(22,30,31,26)(27,28,32,29)(33,34)(35,40)(36,42)(37,41)(38,43)(39,44)(45,46)(47,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,19)(18,21)(20,23)(22,27)(24,25)(26,29)(28,30)(31,32)(33,35,39,37)(34,40,44,41)(36,47,43,45)(38,46,42,48);(2,7)(6,9)(8,12)(14,15)(17,20,24,21)(18,25,23,19)(22,28,31,29)(26,32,30,27)(33,35,39,37)(34,41,44,40)(36,45,43,47)(38,46,42,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,21,24,20)(18,25,23,19)(22,29,31,28)(26,32,30,27)(33,36,39,43)(34,42,44,38)(35,45,37,47)(40,46,41,48);(1,4,5,13)(2,8,7,12)(3,10,11,16)(6,14,9,15)(17,22,24,31)(18,26,23,30)(19,27,25,32)(20,28,21,29)(33,37,39,35)(34,40,44,41)(36,47,43,45)(38,48,42,46);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,13)(15,16)(17,23,24,18)(19,20,25,21)(22,30,31,26)(27,28,32,29)(33,38,39,42)(34,43,44,36)(35,46,37,48)(40,45,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8092095);