#181339

order	512
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#95
commutator	C₂³	C₂²×C₄²
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
socle	C₂²	C₂²×#95
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,b^a=be,e^a=eh,g^a=gi,e^b=ei,f^b=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72235363, C₂×#72210003
maximal subgroup of	#72210000, #72210003, #72235197, …
maximal normal subgroup of	#72210000, #72210003, #72235197, …
maximal quotient of	C₂×#181339, #72240036, #72240809, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88535);