#6372992

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
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	C₂×#3050	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2718 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)(5,6,10,14,11,15,9,24)(7,16,19,26,20,30,18,12)(8,13,23,28,21,27,22,34)(17,29,33,35,31,36,32,25);(5,7,9,18,11,20,10,19)(6,12,14,16,15,26,24,30)(8,17,22,32,21,31,23,33)(13,25,28,29,27,35,34,36);(2,4);(5,8,10,23,11,21,9,22)(6,13,14,28,15,27,24,34)(7,17,19,33,20,31,18,32)(12,25,16,29,26,35,30,36);(8,21)(13,27)(17,31)(22,23)(25,35)(28,34)(29,36)(32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239527, #72239529, #72239536, …
maximal normal subgroup of	#72239527, #72239529, #72239536, …
maximal quotient of	#72309888, #72309927, #72309998, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280188);