#7715505

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	#39131	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4361 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622701);