#427834

order	512
nilpotency class	3
isoclinism	isoclinic class of #417682
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1215 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,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,h^a=hi,c^b=cg,d^b=de,g^b=gi,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, 335030);