#10504568

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	#56610	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10411764);