#8107206

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

4006 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)(3,4)(5,13)(6,7)(8,10)(9,14)(11,16)(12,15)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(36,44)(40,45)(42,48)(46,47);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(34,38)(39,41)(40,45)(46,47);(2,7)(3,8)(9,13)(11,15)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,2)(3,10)(4,8)(5,9)(6,7)(11,16)(12,15)(13,14)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,4,6,10)(2,8,7,3)(5,12,14,16)(9,15,13,11)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=ghi,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,d^c=dgi

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, 8014402);