#10023345

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3942 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(19,25)(23,26)(27,31)(30,32)(33,34,38,43)(35,41,45,39)(36,40,46,48)(37,42,47,44);(2,6)(3,9)(8,11)(10,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,38,45)(34,39,43,41)(36,44,46,42)(37,40,47,48);(2,6)(3,9)(8,11)(10,15)(18,22)(23,26)(24,28)(30,32)(33,36,38,46)(34,40,43,48)(35,44,45,42)(37,39,47,41);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,35)(34,41)(36,44)(37,40)(38,45)(39,43)(42,46)(47,48);(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(33,36)(34,40)(35,44)(37,41)(38,46)(39,47)(42,45)(43,48);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37)(34,42)(35,40)(36,39)(38,47)(41,46)(43,44)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9930541);