#7917236

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4641 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)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,37,40,39);(1,2)(3,7)(4,8)(5,6)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,38)(39,40);(1,3,5,4)(2,7,6,8)(10,15)(11,16)(17,22)(18,23)(26,31)(27,34)(33,36)(35,40);(1,3,5,4)(2,7,6,8)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,37,36,39);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,28)(26,33)(27,35)(29,39)(30,38)(31,36)(32,37)(34,40);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,35,36,40)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=ghi,e^2=gi,f^2=ghi,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, 7824432);