#451891

order	512
nilpotency class	3
isoclinism	isoclinic class of #429607
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2595
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16020
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

765 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 359087);