#8139502

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	#53837	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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)(3,11,14,8)(4,15,13,9)(6,16,12,10)(17,18,21,25)(19,23,26,22)(20,28,29,24)(27,30,32,31)(34,37)(35,36)(38,46)(47,48);(1,3)(2,8)(4,12)(5,11)(6,13)(7,14)(9,10)(15,16)(19,26)(22,23)(27,32)(30,31)(33,34,39,37)(35,40,36,42)(38,41,46,45)(43,47,44,48);(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(1,4,7,13)(2,9,5,15)(3,12,14,6)(8,10,11,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,5,7,2)(3,11,14,8)(4,9,13,15)(6,16,12,10)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,37,39,34)(35,42,36,40)(38,45,46,41)(43,48,44,47);(1,6,7,12)(2,10,5,16)(3,13,14,4)(8,9,11,15)(17,20,21,29)(18,24,25,28)(19,27,26,32)(22,30,23,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8046698);