#7659028

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	#42166	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4520 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7566224);