#8081028

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

4278 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7988224);