#8115417

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3808 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,e^2=g,f^2=gi,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, 8022613);