#6344054

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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3122 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:

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

Polycyclic presentation:

2^9|a^2=g,b^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, #72240121
maximal normal subgroup of	#72235050, #72240121

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251250);