#6384007

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#61030	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2565 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)(3,4)(5,6,10,16,11,17,27,15)(7,18,22,29,23,34,21,12)(8,13,26,28,20,31,25,35)(9,14,19,32,24,33,36,30);(5,7)(6,12)(8,24)(9,20)(10,22)(11,23)(13,30)(14,28)(15,34)(16,18)(17,29)(19,25)(21,27)(26,36)(31,32)(33,35);(2,4);(5,8)(6,13)(7,19)(9,21)(10,26)(11,20)(12,14)(15,35)(16,28)(17,31)(18,32)(22,24)(23,36)(25,27)(29,33)(30,34);(5,9,10,19,11,24,27,36)(6,14,16,32,17,33,15,30)(7,20,22,25,23,8,21,26)(12,28,18,31,29,35,34,13)

Polycyclic presentation:

2^9|a^2=h,e^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=df,f^b=fi,f^d=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2955, #72320787
maximal normal subgroup of	D₄×#2955, #72320787
maximal quotient of	#72286208, #72322026, #72322056

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291203);