#8746101

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	C₂×#555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4125 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)(3,7)(4,8)(5,6)(9,10,15,12)(11,16,13,17)(14,18,22,20)(19,23,21,24)(25,26,30,34)(27,35,36,31)(28,32,38,40)(29,33,37,39);(2,6)(3,8)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,27,30,36)(26,31,34,35)(28,37,38,29)(32,33,40,39);(1,3)(2,4)(5,8)(6,7)(9,11,15,13)(10,17,12,16)(14,19,22,21)(18,24,20,23)(25,27,30,36)(26,31,34,35)(28,37,38,29)(32,33,40,39);(1,3)(2,4)(5,8)(6,7)(9,12,15,10)(11,16,13,17)(14,20,22,18)(19,23,21,24)(25,28,30,38)(26,32,34,40)(27,29,36,37)(31,39,35,33);(1,4)(2,3)(5,7)(6,8)(9,13,15,11)(10,16,12,17)(14,21,22,19)(18,23,20,24)(25,26)(27,31)(28,32)(29,39)(30,34)(33,37)(35,36)(38,40);(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,29)(26,33)(27,28)(30,37)(31,40)(32,35)(34,39)(36,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=hi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8653297);