#455248

order	512
nilpotency class	3
isoclinism	isoclinic class of #455155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1279 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362444);