#7763303

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	#34624	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4402 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)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26)(27,29)(28,36)(30,33)(31,35)(32,39)(34,40)(37,38);(1,3,5,8)(2,6,4,7)(25,27,30,35)(26,31,33,29)(28,37,39,34)(32,38,36,40);(1,4,5,2)(3,7,8,6)(10,15)(11,16)(17,22)(18,23)(25,26,30,33)(27,29,35,31)(28,32,39,36)(34,38,37,40);(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,26)(27,29)(28,32)(30,33)(31,35)(34,38)(36,39)(37,40);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,28,30,39)(26,32,33,36)(27,34,35,37)(29,38,31,40);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,29)(26,27)(28,38)(30,31)(32,34)(33,35)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=hi,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi

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, 7670499);