#116248

order	512
nilpotency class	3
isoclinism	isoclinic class of #10382
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1439
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,f^2=hi,g^2=hi,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,f^b=fi,d^c=dh,e^c=ehi,f^c=fhi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266586, #72266644
maximal normal subgroup of	#72266586, #72266644
maximal quotient of	#72266589

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23444);