#374453

order	512
nilpotency class	3
isoclinism	isoclinic class of #24808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1686 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fhi,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,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, 281649);