#213334

order	512
nilpotency class	3
isoclinism	isoclinic class of #19671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#127
prime radical	#19619	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

717 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,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=ci,d^b=dh,d^c=dh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120530);