#6343674

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343394
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62812	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,e^2=hi,f^2=i,b^a=bf,c^a=ch,e^a=ehi,f^a=fh,f^b=fi,g^b=ghi,d^c=dhi

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, 6250870);