#6373783

order	512
nilpotency class	3
isoclinism	isoclinic class of #60460
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ci,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280979);