#122176

order	512
nilpotency class	3
isoclinism	isoclinic class of #122176
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

768 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)(3,6,10,14)(4,11)(5,8)(7,12)(9,13,16,15)(17,18)(19,26)(20,29)(21,25)(22,27)(23,28)(24,30)(31,32);(1,3)(2,6,8,14)(4,9)(5,10)(7,15,11,13)(12,16)(17,19)(18,22,25,26)(20,28)(21,27)(23,29,31,24)(30,32);(1,4)(2,7,8,11)(3,9)(5,12)(6,13,14,15)(10,16)(17,19)(18,23,25,31)(20,28)(21,27)(22,24,26,29)(30,32)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,e^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,e^c=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29372);