#8075376

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	#37824	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4327 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=g,e^2=h,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, 7982572);