#405578

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	#23406	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1326 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)(3,6)(4,5)(7,8)(9,10)(11,23)(12,28)(13,31)(14,21)(15,20)(16,22)(17,26)(18,30)(19,32)(24,35)(25,40)(27,39)(29,36)(33,38)(34,37);(2,5)(6,8)(9,11,14,26,16,27,15,29)(10,17,20,23,22,36,21,39)(12,24,30,13,25,32,34,33)(18,19,28,31,37,35,40,38);(9,12)(10,18)(11,24)(13,26)(14,30)(15,34)(16,25)(17,35)(19,36)(20,28)(21,40)(22,37)(23,38)(27,32)(29,33)(31,39);(1,3,4,7)(2,6,5,8)(9,13,14,32,16,33,15,24)(10,19,20,31,22,35,21,38)(11,25,26,34,27,12,29,30)(17,28,23,37,36,40,39,18)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264612
maximal normal subgroup of	#72264612

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312774);