#8062875

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	#45997	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4371 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7970071);