#8031595

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	#59091	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4351 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=ghi,e^2=gi,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, 7938791);