#361636

order	512
nilpotency class	3
isoclinism	isoclinic class of #184991
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3500	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3666 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317916, #72317949, #72317961, …
maximal normal subgroup of	#72317916, #72317949, #72317961, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268832);