#8097031

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

4092 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004227);