#9945511

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	#55218	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9852707);