#7893676

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

4113 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=hi,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, 7800872);