#8902532

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	#54884	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4123 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,15)(13,14)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(33,34,38,40)(35,39,42,37)(36,41,45,47)(43,44,48,46);(1,3,7,4)(2,8,6,9)(5,12,14,11)(10,16,13,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,35)(34,39)(36,43)(37,40)(38,42)(41,44)(45,48)(46,47);(2,6)(8,9)(10,13)(15,16)(21,29)(24,31)(26,32)(28,30)(34,40)(35,42)(41,47)(43,48);(1,4,7,3)(2,9,6,8)(5,11,14,12)(10,15,13,16)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,36,38,45)(34,41,40,47)(35,43,42,48)(37,46,39,44);(1,5,7,14)(2,10,6,13)(3,11,4,12)(8,15,9,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,37)(34,35)(36,44)(38,39)(40,42)(41,48)(43,47)(45,46);(1,6,7,2)(3,9,4,8)(5,13,14,10)(11,16,12,15)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,35)(34,37)(36,43)(38,42)(39,40)(41,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=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, 8809728);