#455249

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×#3062	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1335 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=ei,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, 362445);