#6344055

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,e^2=i,f^2=i,g^2=hi,b^a=bf,c^a=ci,d^a=dh,f^a=fh,f^b=fi,g^b=ghi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235050, #72239082, #72239086, …
maximal normal subgroup of	#72235050, #72239082, #72239086, …
maximal quotient of	#72239121, #72239128

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251251);