#201669

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	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1886 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)(3,7,12,20)(4,14,15,8)(5,9)(6,10)(11,19,25,32)(13,21,27,30)(16,22,29,23)(17,18)(24,31,26,28);(1,3,6,13)(2,7,18,30)(4,11,15,26)(5,12,17,27)(8,19,22,32)(9,20,10,21)(14,28,23,31)(16,24,29,25);(1,4,5,16)(2,8,9,23)(3,11,12,24)(6,15,17,29)(7,19,20,31)(10,14,18,22)(13,26,27,25)(21,28,30,32);(3,12)(7,20)(11,24)(13,27)(19,31)(21,30)(25,26)(28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=h,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	#72238044, #72238048, #72238050, …
maximal normal subgroup of	#72238044, #72238048, #72238050, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108865);