#8434356

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	#37903	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4185 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8341552);