#369874

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1125 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=g,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=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, 277070);