#8075511

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	#43902	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4357 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=ei,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, 7982707);