#381495

order	512
nilpotency class	3
isoclinism	isoclinic class of #29610
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1215 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)(3,6)(4,5)(7,8)(9,10,16,21,14,20,15,38)(11,22,27,30,25,17,26,34)(12,28,32,18,31,39,33,24)(13,19,37,23,35,29,36,40);(2,5)(6,8)(9,11)(10,17)(12,29)(13,24)(14,25)(15,26)(16,27)(18,35)(19,31)(20,22)(21,34)(23,33)(28,37)(30,38)(32,40)(36,39);(1,3,4,7)(2,6,5,8)(9,12,14,31)(10,18,20,24)(11,23,25,40)(13,34,35,30)(15,33,16,32)(17,37,22,36)(19,27,29,26)(21,39,38,28);(3,7)(6,8)(9,13,16,37,14,35,15,36)(10,19,21,23,20,29,38,40)(11,24,27,28,25,18,26,39)(12,30,32,17,31,34,33,22)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288691);