#411006

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

704 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=gh,c^2=g,d^2=e,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	#72265390, #72307328, #72307337
maximal normal subgroup of	#72265390, #72307328, #72307337

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318202);