#8114396

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	#54575	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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)(3,11)(4,15)(5,7)(6,16)(8,14)(9,13)(10,12)(18,23)(19,26)(20,28)(29,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,3,7,14)(2,8,5,11)(4,6,13,12)(9,16,15,10)(17,18,22,23)(19,21,26,24)(20,25,28,30)(27,29,32,31)(34,38)(39,41)(40,45)(46,47);(1,2,7,5)(3,8,14,11)(4,15,13,9)(6,10,12,16)(17,19,22,26)(18,24,23,21)(20,27,28,32)(25,31,30,29)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4,7,13)(2,9,5,15)(3,12,14,6)(8,10,11,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,7,2)(3,8,14,11)(4,15,13,9)(6,16,12,10)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32);(1,6,7,12)(2,10,5,16)(3,13,14,4)(8,9,11,15)(17,21,22,24)(18,26,23,19)(20,29,28,31)(25,32,30,27)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=gi,e^2=gi,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,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8021592);