#8285458

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4186 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:

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

Polycyclic presentation:

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

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, 8192654);