#6343250

order	512
nilpotency class	3
isoclinism	isoclinic class of #62295
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62816	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3009 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:

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

Polycyclic presentation:

2^9|a^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,e^a=ehi,f^a=fh,e^b=ei,f^b=fh,d^c=di

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250446);