#8854291

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	#35053	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) = 128

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

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

Polycyclic presentation:

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

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, 8761487);