#359104

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224916, #72280138, #72313487
maximal normal subgroup of	#72224916, #72280138, #72313487
maximal quotient of	#72224914

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266300);