#6370097

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	C₂×#575	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2990 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#60182, #72224755, #72224756, …
maximal normal subgroup of	C₄×#60182, #72224755, #72224756, …
maximal quotient of	#72280039

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277293);