#8907953

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	#56311	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4271 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)(3,10)(4,5)(6,7)(8,11)(9,15)(12,13)(14,16)(17,18,22,27)(19,25,30,23)(20,21,26,24)(28,32,31,29)(33,34)(35,40)(36,41)(37,42)(38,39)(43,45)(44,48)(46,47);(2,7)(4,9)(8,10)(12,14)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,27)(26,31)(33,35)(34,40)(36,44)(37,46)(38,45)(39,43)(41,48)(42,47);(2,7)(3,11)(5,15)(12,14)(20,26)(21,24)(28,31)(29,32)(34,38)(35,43)(36,46)(42,48);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,20,22,26)(18,24,27,21)(19,28,30,31)(23,32,25,29)(33,36,39,46)(34,41,38,47)(35,44,43,37)(40,48,45,42);(1,4,6,9)(2,5,7,15)(3,12,11,14)(8,16,10,13)(17,19)(18,25)(20,31)(21,29)(22,30)(23,27)(24,32)(26,28)(33,37,39,44)(34,42,38,48)(35,36,43,46)(40,41,45,47);(1,5)(2,9)(3,13)(4,7)(6,15)(8,12)(10,14)(11,16)(17,21,22,24)(18,26,27,20)(19,29,30,32)(23,31,25,28)(33,38,39,34)(35,45,43,40)(36,47,46,41)(37,48,44,42)

Polycyclic presentation:

2^9|a^2=ghi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815149);