#8528132

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	#39813	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4302 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8435328);