#7673630

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4322 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)(3,6)(4,5)(7,8)(9,10,13,12)(11,15,16,14)(17,18,22,23)(19,24,27,28)(20,25,33,31)(21,34,35,26)(29,36,39,41)(30,42,40,37)(32,44,45,38)(43,48,47,46);(9,11,13,16)(10,14,12,15)(18,23)(19,27)(25,31)(26,34)(29,39)(30,40)(38,44)(43,47);(2,5)(6,8)(17,18)(19,28)(20,31)(21,26)(22,23)(24,27)(25,33)(29,36)(30,42)(32,44)(34,35)(37,40)(38,45)(39,41)(43,46)(47,48);(1,2)(3,6)(4,5)(7,8)(9,10,13,12)(11,14,16,15)(17,19)(18,24)(20,29)(21,30)(22,27)(23,28)(25,36)(26,37)(31,41)(32,43)(33,39)(34,42)(35,40)(38,46)(44,48)(45,47);(9,10,13,12)(11,14,16,15)(17,20,22,33)(18,25,23,31)(19,29,27,39)(21,32,35,45)(24,36,28,41)(26,38,34,44)(30,43,40,47)(37,46,42,48);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,21)(18,26)(19,30)(20,32)(22,35)(23,34)(24,37)(25,38)(27,40)(28,42)(29,43)(31,44)(33,45)(36,46)(39,47)(41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7580826);