#390428

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	#23386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=egh,e^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 297624);