#7981198

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	#39722	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4500 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7888394);