#393208

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1142 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265145, #72265153, #72265222, …
maximal normal subgroup of	#72265145, #72265153, #72265222, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300404);