#8097775

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3894 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)(3,8)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,18)(19,24)(20,23)(21,25)(22,27)(26,30)(28,31)(29,32)(34,39)(40,46)(41,42)(47,48);(4,13)(7,12)(10,16)(14,15)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(34,39)(35,43)(41,42)(44,45);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,20,21,28)(18,23,25,31)(19,26,27,32)(22,29,24,30)(33,34)(35,40)(36,41)(37,42)(38,39)(43,46)(44,47)(45,48);(1,4)(2,7)(3,10)(5,13)(6,14)(8,15)(9,12)(11,16)(33,35)(34,40)(36,44)(37,45)(38,43)(39,46)(41,47)(42,48);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,19)(18,24)(20,26)(21,27)(22,25)(23,30)(28,32)(29,31)(33,36,38,37)(34,41,39,42)(35,44,43,45)(40,47,46,48);(1,2)(3,6)(4,7)(5,9)(8,11)(10,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,25)(24,27)(26,29)(28,31)(30,32)(33,37,38,36)(34,42,39,41)(35,45,43,44)(40,48,46,47)

Polycyclic presentation:

2^9|c^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi

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, 8004971);