#201821

order	512
nilpotency class	3
isoclinism	isoclinic class of #201664
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278098, #72278163, #72278199
maximal normal subgroup of	#72278098, #72278163, #72278199

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109017);