#236645

order	512
nilpotency class	3
isoclinism	isoclinic class of #236644
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3153	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|c^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=cg,e^b=eh,f^b=fi,g^b=gh,d^c=dhi,e^c=ei,f^c=fhi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 143841);