#8096857

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004053);