#8374178

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	#58961	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4066 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=h,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8281374);