#8434768

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	#35009	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4448 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8341964);