#7918778

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=h,e^2=hi,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, 7825974);