#8933096

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	#35422	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4187 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

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, 8840292);