#429995

order	512
nilpotency class	3
isoclinism	isoclinic class of #419461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,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, 337191);