#116015

order	512
nilpotency class	3
isoclinism	isoclinic class of #116007
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
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

581 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,e^b=ei,d^c=dh,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, 23211);