#107315

order	512
nilpotency class	3
isoclinism	isoclinic class of #107314
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

976 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)(3,7)(4,17,20,26)(5,14)(6,13)(8,15)(9,16)(10,18)(11,19,27,24)(12,23)(21,30,25,28)(22,29,32,31);(1,3,7,16,8,13,6,18)(2,10,14,5,15,9,12,23)(4,17,19,29,20,26,24,31)(11,25,28,22,27,21,30,32);(1,4)(2,11)(3,17)(5,22)(6,24)(7,19)(8,20)(9,21)(10,25)(12,30)(13,26)(14,28)(15,27)(16,29)(18,31)(23,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di,e^b=ehi,d^c=dhi,e^c=eh,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14511);