#8108726

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	#53241	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015922);