#9909955

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	#51937	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4166 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)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18)(19,22)(20,27)(21,23)(24,28)(25,31)(26,29)(30,32)(33,34,38,39)(35,41,43,36)(37,40,46,47)(42,48,45,44);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,21,26)(18,22,23,29)(20,25,28,32)(24,30,27,31)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,46)(47,48);(2,6)(7,9)(8,12)(14,15)(18,23)(22,29)(24,27)(30,31)(34,39)(36,41)(40,47)(44,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,18)(19,22)(20,24)(21,23)(25,30)(26,29)(27,28)(31,32)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,47)(46,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,37,38,46)(34,40,39,47)(35,42,43,45)(36,44,41,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,20)(18,24)(19,25)(21,28)(22,30)(23,27)(26,32)(29,31)(33,35)(34,36)(37,45)(38,43)(39,41)(40,48)(42,46)(44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817151);