#10171641

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3925 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=gi,e^2=gi,f^2=g,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, 10078837);