#8334542

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	#53697	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3985 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=i,e^2=h,f^2=hi,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, 8241738);