#441115

order	512
nilpotency class	3
isoclinism	isoclinic class of #420453
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1422 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=ei,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,d^b=de,f^b=fi,h^b=hi,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285701, #72285809, #72286206
maximal normal subgroup of	#72285701, #72285809, #72286206

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348311);