#8810079

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	#48281	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8717275);