#8720541

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4001 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)(3,8)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,18,23,25)(19,27,21,24)(20,22,28,26)(29,30,31,32)(33,34,38,37)(35,40,42,39)(36,44,45,41)(43,46,48,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,23,21)(18,24,25,27)(20,31,28,29)(22,30,26,32)(35,42)(39,40)(43,48)(46,47);(4,13)(7,12)(10,16)(14,15)(18,25)(20,28)(24,27)(29,31)(33,35,38,42)(34,39,37,40)(36,43,45,48)(41,46,44,47);(1,4,5,13)(2,7,9,12)(3,10,11,16)(6,14,8,15)(17,20)(18,26)(19,29)(21,31)(22,25)(23,28)(24,30)(27,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,46)(45,48);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,21,23,19)(18,27,25,24)(20,31,28,29)(22,30,26,32)(33,36,38,45)(34,41,37,44)(35,43,42,48)(39,46,40,47);(1,2)(3,6)(4,12)(5,9)(7,13)(8,11)(10,15)(14,16)(17,22,23,26)(18,28,25,20)(19,30,21,32)(24,29,27,31)(33,37,38,34)(35,40,42,39)(36,41,45,44)(43,46,48,47)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627737);