#115329

order	512
nilpotency class	3
isoclinism	isoclinic class of #115318
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

871 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 22525);