#8701504

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	#45602	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 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)(3,13)(4,10)(5,15)(6,11)(7,9)(8,12)(14,16)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,31)(29,32);(1,3,8,7)(2,9,12,13)(4,14,5,11)(6,15,16,10)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,30)(31,32)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(1,4,8,5)(2,10,12,15)(3,14,7,11)(6,13,16,9)(17,19)(18,21)(20,27)(22,25)(23,26)(24,31)(28,29)(30,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(1,5,8,4)(2,10,12,15)(3,14,7,11)(6,9,16,13)(17,19)(18,23)(20,27)(21,26)(22,25)(24,29)(28,31)(30,32)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,6,8,16)(2,11,12,14)(3,15,7,10)(4,13,5,9)(17,20,22,30)(18,24,26,28)(19,27,25,32)(21,31,23,29)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,46)(45,48);(1,7,8,3)(2,9,12,13)(4,11,5,14)(6,15,16,10)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,28,32)(33,37,38,34)(35,41,43,39)(36,40,45,44)(42,46,48,47)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8608700);