#181789

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1548 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)(3,11)(4,8)(5,9)(6,10)(7,12)(13,16)(14,15)(17,18,20,23,21,24,28,31)(19,25,27,32,26,22,29,30);(1,3)(2,7,10,16)(4,13)(5,12)(6,14)(8,11,9,15)(17,19)(18,22,24,25)(20,27)(21,26)(23,30,31,32)(28,29);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11);(1,4,6,5)(2,9,10,8)(3,13,14,12)(7,15,16,11)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=h,f^2=i,b^a=be,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224694, #72224697, #72224699, …
maximal normal subgroup of	#72224694, #72224697, #72224699, …
maximal quotient of	#72239282

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88985);