#119476

order	512
nilpotency class	3
isoclinism	isoclinic class of #119474
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

776 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)(3,11,12,15)(4,7)(5,9)(6,10)(8,14,16,13)(17,18)(19,22)(20,27,28,23)(21,24)(25,31,32,29)(26,30);(2,7)(3,12)(4,5)(8,15)(9,10)(11,16)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31);(1,3,4,13,6,12,5,14)(2,8,9,11,10,16,7,15)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^c=fh,g^c=ghi

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, 26672);