#7648303

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4538 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7555499);