#8902462

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	#58961	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4082 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=hi,e^2=hi,f^2=h,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, 8809658);