#116008

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	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

909 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,f^2=h,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, 23204);