#8127532

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8034728);