#6369908

order	512
nilpotency class	3
isoclinism	isoclinic class of #3585
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,f^2=i,b^a=bf,e^a=ei,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#60182, #72224750, #72226672
maximal normal subgroup of	C₄×#60182, #72224750, #72226672

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277104);