#8552145

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	#53743	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8459341);