#211796

order	512
nilpotency class	3
isoclinism	isoclinic class of #19162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#121
prime radical	#19133	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

719 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)(3,5)(4,7)(6,8)(9,10,14,18,15,19,13,29)(11,20,24,30,21,35,23,16)(12,17,27,33,28,34,26,40)(22,36,38,32,25,39,37,31);(9,11,13,23,15,21,14,24)(10,16,18,20,19,30,29,35)(12,25,26,38,28,22,27,37)(17,32,33,39,34,31,40,36);(2,5)(3,8)(6,7)(11,21)(16,30)(20,35)(22,25)(23,24)(31,32)(36,39)(37,38);(9,12,14,27,15,28,13,26)(10,17,18,33,19,34,29,40)(11,22,24,38,21,25,23,37)(16,31,20,36,30,32,35,39)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=g,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 118992);