#205618

order	512
nilpotency class	3
isoclinism	isoclinic class of #205569
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60700	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,e^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305458, #72305464, #72305488
maximal normal subgroup of	#72305458, #72305464, #72305488

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112814);