#8118779

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	#38174	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4137 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)(3,4)(5,13)(6,7)(8,10)(9,14)(11,15)(12,16)(21,29)(24,31)(27,28)(30,32)(33,34,39,41)(35,42,38,40)(36,37,45,47)(43,46,48,44);(1,3)(2,4)(5,11)(6,10)(7,8)(9,12)(13,15)(14,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,35,39,38)(34,40,41,42)(36,43,45,48)(37,44,47,46);(2,7)(3,10)(9,13)(11,16)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,36)(34,37)(35,43)(38,48)(39,45)(40,44)(41,47)(42,46);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,37)(34,36)(35,44)(38,46)(39,47)(40,43)(41,45)(42,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,13)(14,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,38,39,35)(34,40,41,42)(36,43,45,48)(37,46,47,44);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,36)(35,44)(38,46)(39,47)(40,43)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8025975);