#441114

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

1494 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)(11,15,21,29)(12,16,23,32)(13,18)(14,17)(19,25)(20,30)(22,34,26,28)(24,31,37,33)(27,35)(36,39)(38,40);(1,2,5,6)(3,7,4,8)(9,11,13,21)(10,15,18,29)(12,20,23,35)(14,26,25,22)(16,30,32,27)(17,28,19,34)(24,38,37,36)(31,40,33,39);(1,3,5,4)(2,7,6,8)(9,12,13,23)(10,16)(11,20,21,35)(14,24,25,37)(15,27)(17,31)(18,32)(19,33)(22,36,26,38)(28,39)(29,30)(34,40);(1,4,5,3)(2,7,6,8)(10,17,18,19)(11,21)(15,28,29,34)(16,31,32,33)(20,35)(22,26)(27,39,30,40)(36,38)

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=e,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	#72285193, #72285208, #72285237, …
maximal normal subgroup of	#72285193, #72285208, #72285237, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348310);