#441153

order	512
nilpotency class	3
isoclinism	isoclinic class of #420420
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	#60456	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348349);