#8114419

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	#54964	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3901 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)(3,5,4,9)(6,14,15,10)(11,16,12,13)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(34,36)(38,46)(40,43)(42,45);(1,3)(2,5)(4,7)(6,11)(8,9)(10,13)(12,15)(14,16)(19,25)(22,28)(26,31)(29,32)(33,34,39,36)(35,40,37,43)(38,41,46,44)(42,47,45,48);(2,8)(3,4)(10,14)(11,12)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,35,39,37)(34,40,36,43)(38,42,46,45)(41,47,44,48);(1,4,7,3)(2,9,8,5)(6,12,15,11)(10,16,14,13)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,36,39,34)(35,40,37,43)(38,44,46,41)(42,47,45,48);(1,5,7,9)(2,3,8,4)(6,13,15,16)(10,11,14,12)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,37,39,35)(34,40,36,43)(38,45,46,42)(41,47,44,48);(1,6,7,15)(2,10,8,14)(3,11,4,12)(5,13,9,16)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=gi,e^2=gi,f^2=h,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, 8021615);