#417963

order	512
nilpotency class	3
isoclinism	isoclinic class of #417566
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2002 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)(5,6,9,11)(7,15)(8,13,19,25)(10,22,21,14)(12,18)(16,28)(17,31)(20,30,24,29)(23,26)(27,34)(32,36)(33,35);(6,11)(7,16,18,23)(8,19)(12,26,15,28)(14,22)(17,32,34,33)(20,24)(27,35,31,36);(5,7)(6,12)(8,17)(9,18)(10,23)(11,15)(13,27)(14,26)(16,21)(19,34)(20,32)(22,28)(24,33)(25,31)(29,36)(30,35);(2,4)(5,8,9,19)(6,13,11,25)(7,17,18,34)(10,24,21,20)(12,27,15,31)(14,30,22,29)(16,32,23,33)(26,35,28,36)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,d^2=i,g^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,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, 325159);