#8001021

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4501 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7908217);