#8075308

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	C₂×H(3,𝔽₄)	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4453 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^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, 7982504);