#7992771

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	#37536	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7899967);