#6381904

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60422	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2511 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,g^c=gi,e^d=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320787, #72343918
maximal normal subgroup of	#72320787, #72343918
maximal quotient of	#72343943, #72343950

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289100);