#8690875

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	#44001	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4611 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)(11,14)(12,17)(13,22)(15,18)(16,20)(19,21)(23,24)(25,26,30,35)(27,31,37,40)(28,33,38,32)(29,36,39,34);(3,7)(6,8)(9,11)(10,16)(12,19)(13,23)(14,18)(15,20)(17,24)(21,22)(25,27)(26,31)(28,34)(29,33)(30,37)(32,39)(35,40)(36,38);(1,2)(3,6)(4,5)(7,8)(9,12,15,13)(10,17,18,22)(11,19,20,23)(14,21,16,24)(25,28)(26,32)(27,36)(29,31)(30,38)(33,35)(34,37)(39,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,12)(10,17,18,22)(11,19,20,23)(14,24,16,21)(25,28)(26,33)(27,34)(29,31)(30,38)(32,35)(36,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,14,15,16)(10,11,18,20)(12,21,13,24)(17,19,22,23)(25,29)(26,34)(27,33)(28,31)(30,39)(32,37)(35,36)(38,40);(9,11)(10,14)(12,19)(13,23)(15,20)(16,18)(17,21)(22,24)(25,27)(26,31)(28,36)(29,32)(30,37)(33,39)(34,38)(35,40)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8598071);