#9230862

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	#40053	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4193 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9138058);