#115393

order	512
nilpotency class	3
isoclinism	isoclinic class of #115392
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

882 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)(3,9,10,15)(4,7)(5,8)(6,11,14,12)(13,16)(17,18,21,27)(19,23)(20,28)(22,25,31,26)(24,32)(29,30);(6,14)(9,15)(17,19)(18,23)(20,27)(21,28)(22,24)(25,32)(26,30)(29,31);(1,3)(2,6)(4,12)(5,10)(7,15)(8,14)(9,16)(11,13)(17,20,21,23)(18,24,27,29)(19,26,28,25)(22,32,31,30)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22589);