#6344059

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62812	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3869 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:

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

Polycyclic presentation:

2^9|a^2=g,e^2=h,f^2=i,g^2=hi,b^a=bf,c^a=ci,f^a=fh,f^b=fi,g^b=ghi,d^c=dh

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, 6251255);