#406792

order	512
nilpotency class	3
isoclinism	isoclinic class of #393988
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1122 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307083, #72307337, #72307382
maximal normal subgroup of	#72307083, #72307337, #72307382

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313988);