#8341622

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3997 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=h,e^2=hi,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, 8248818);