#10195447

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	#49405	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 10102643);