#373216

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1007 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)(5,6,11,15,9,13,10,22)(7,12,16,24,18,26,19,30)(8,14,21,29,20,27,23,28)(17,32,33,35,34,25,31,36);(1,3,4,2)(5,7,11,19,9,18,10,16)(6,12,22,24,13,26,15,30)(8,17,23,33,20,34,21,31)(14,25,29,35,27,32,28,36);(6,13)(7,16,18,19)(12,24,26,30)(14,27)(15,22)(17,31,34,33)(25,35,32,36)(28,29);(5,8,11,23,9,20,10,21)(6,14,22,29,13,27,15,28)(7,17,19,33,18,34,16,31)(12,25,24,35,26,32,30,36)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fi,d^2=fg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^d=ei,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, 280412);