#9712854

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3889 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9620050);