#364923

order	512
nilpotency class	3
isoclinism	isoclinic class of #23903
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	#34633	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1724 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)(5,6,9,14,10,13,20,27)(7,11,17,23,16,22,30,35)(8,12,19,26,18,25,31,33)(15,24,29,36,28,21,32,34);(1,2,4,3)(5,7,9,17,10,16,20,30)(6,11,14,23,13,22,27,35)(8,15,19,29,18,28,31,32)(12,24,26,36,25,21,33,34);(5,8)(6,12)(7,15)(9,19)(10,18)(11,21)(13,25)(14,26)(16,28)(17,29)(20,31)(22,24)(23,34)(27,33)(30,32)(35,36);(1,3,4,2)(8,18)(12,25)(15,28)(19,31)(21,24)(26,33)(29,32)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303578, #72303579, #72303582
maximal normal subgroup of	#72303578, #72303579, #72303582

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272119);