#7920028

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	#35752	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4361 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=gi,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7827224);