#374439

order	512
nilpotency class	3
isoclinism	isoclinic class of #27987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1682 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281635);