#8107297

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	#40999	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4074 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=ghi,e^2=gh,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, 8014493);