#8095401

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	#45515	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4253 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002597);