#8097797

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	#52985	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 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:

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

Polycyclic presentation:

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

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, 8004993);