#8734795

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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)(3,7)(4,13)(5,9)(6,10)(8,12)(11,16)(14,15)(17,18,23,25)(19,24,28,32)(20,21,30,26)(22,27,31,29)(33,34,39,40)(35,43,37,41)(36,42,47,46)(38,44,48,45);(1,3,6,11)(2,7,10,16)(4,5,12,15)(8,14,13,9)(17,19)(18,24)(20,31)(21,29)(22,30)(23,28)(25,32)(26,27)(34,40)(36,47)(41,43)(44,45);(3,11)(4,12)(7,16)(8,13)(18,25)(19,28)(20,30)(27,29)(33,35,39,37)(34,41,40,43)(36,44,47,45)(38,46,48,42);(1,4)(2,8)(3,5)(6,12)(7,14)(9,16)(10,13)(11,15)(17,20)(18,26)(19,22)(21,25)(23,30)(24,29)(27,32)(28,31)(33,36,39,47)(34,42,40,46)(35,45,37,44)(38,41,48,43);(1,2)(3,7)(4,8)(5,14)(6,10)(9,15)(11,16)(12,13)(17,21,23,26)(18,20,25,30)(19,29,28,27)(22,24,31,32)(33,37,39,35)(34,43,40,41)(36,44,47,45)(38,46,48,42);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,14)(13,16)(17,22)(18,27)(19,30)(20,28)(21,32)(23,31)(24,26)(25,29)(33,38)(34,44)(35,46)(36,43)(37,42)(39,48)(40,45)(41,47)

Polycyclic presentation:

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

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, 8641991);