#6343928

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343584
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#3500	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3884 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281493, #72281505, #72281526, …
maximal normal subgroup of	#72281493, #72281505, #72281526, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251124);