#204431

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1183 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh,g^b=gh,d^c=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111627);