#6371886

order	512
nilpotency class	3
isoclinism	isoclinic class of #60460
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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2570 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)(5,6,10,16,11,17,25,15)(7,12,8,20,19,13,22,27)(9,14,24,31,23,29,35,30)(18,26,21,33,32,28,34,36);(2,4)(5,7,10,8,11,19,25,22)(6,12,16,20,17,13,15,27)(9,18,24,21,23,32,35,34)(14,26,31,33,29,28,30,36);(5,8,10,19,11,22,25,7)(6,13,15,20,17,12,16,27)(9,21,24,32,23,34,35,18)(14,28,30,33,29,26,31,36);(5,9)(6,14)(7,18)(8,21)(10,24)(11,23)(12,26)(13,28)(15,30)(16,31)(17,29)(19,32)(20,33)(22,34)(25,35)(27,36);(9,23)(14,29)(18,32)(21,34)(24,35)(26,28)(30,31)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264929, #72264946, #72265051, …
maximal normal subgroup of	#72264929, #72264946, #72265051, …
maximal quotient of	D₄×#1946

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279082);