#427947

order	512
nilpotency class	3
isoclinism	isoclinic class of #417728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1321 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)(7,17,21,29)(8,22,24,30)(9,12)(10,15)(11,16)(13,19,26,20)(14,25,27,23)(18,32)(28,33)(31,34)(35,36);(6,12)(10,11)(13,26)(14,27)(18,33)(19,20)(23,25)(31,36);(5,7)(6,13)(8,18)(9,21)(10,20)(11,19)(12,26)(14,28)(15,29)(16,17)(22,36)(23,35)(24,33)(25,34)(27,32)(30,31);(2,4)(5,8,11,23,9,24,10,25)(6,14,16,30,12,27,15,22)(7,18,19,35,21,33,20,34)(13,28,17,31,26,32,29,36)

Polycyclic presentation:

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

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, 335143);