#10147568

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3932 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)(18,23)(19,26)(21,29)(28,31)(33,34,39,44)(35,43,38,40)(36,37,42,41)(45,46,47,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,13)(12,15)(14,16)(17,18,22,23)(19,20,26,24)(21,25,29,30)(27,28,32,31)(33,35,39,38)(34,40,44,43)(36,45,42,47)(37,48,41,46);(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28)(33,36,39,42)(34,41,44,37)(35,45,38,47)(40,46,43,48);(1,3,6,10)(2,8,7,5)(4,12,14,11)(9,15,16,13)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,36)(34,41)(35,45)(37,44)(38,47)(39,42)(40,46)(43,48);(1,4,6,14)(2,9,7,16)(3,11,10,12)(5,15,8,13)(17,20)(18,19)(21,28)(22,24)(23,26)(25,27)(29,31)(30,32)(33,37,39,41)(34,42,44,36)(35,46,38,48)(40,45,43,47);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,11,16,12)(17,21,22,29)(18,25,23,30)(19,27,26,32)(20,28,24,31)(33,38,39,35)(34,43,44,40)(36,47,42,45)(37,46,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,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, 10054764);