#441116

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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1478 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)(11,15,21,31)(12,23,25,16)(13,19)(14,17)(18,27)(20,29)(22,34,28,30)(24,36)(26,33,38,32)(35,40)(37,39);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,24)(13,21)(14,28)(16,29)(17,30)(18,34)(19,31)(20,25)(22,27)(23,36)(26,35)(32,39)(33,40)(37,38);(1,3,5,4)(2,7,6,8)(9,12)(10,16,19,23)(11,20)(13,25)(14,26)(15,29,31,36)(17,32,18,33)(21,24)(22,35)(27,38)(28,37)(30,39,34,40);(1,4,5,3)(2,7,6,8)(10,17,19,18)(11,21)(15,30,31,34)(16,32,23,33)(20,24)(22,28)(29,39,36,40)(35,37)

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=ei,d^2=ei,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	#72285763, #72285809, #72286134, …
maximal normal subgroup of	#72285763, #72285809, #72286134, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348312);