#111144

order	512
nilpotency class	3
isoclinism	isoclinic class of #9183
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

583 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,6,9,7,3,5,11)(4,8,13,15,14,10,12,16)(17,18,21,27,23,26,22,35)(19,24,30,38,29,37,34,39)(20,25,32,40,31,41,33,46)(28,42,44,48,43,36,45,47);(1,3,7,2)(4,10,14,8)(5,9,6,11)(12,15,13,16)(17,19,22,30,23,29,21,34)(18,24,27,39,26,37,35,38)(20,28,33,44,31,43,32,45)(25,36,40,48,41,42,46,47);(1,4,6,13,7,14,5,12)(2,8,9,15,3,10,11,16)(17,20,22,33,23,31,21,32)(18,25,35,46,26,41,27,40)(19,28,30,44,29,43,34,45)(24,36,38,47,37,42,39,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239870
maximal normal subgroup of	#72239870
maximal quotient of	#72239871

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18340);