#8096917

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

3962 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^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, 8004113);