#349162

order	512
nilpotency class	3
isoclinism	isoclinic class of #349150
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	D₄×#41	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,d^b=di,f^c=fi,g^d=gh

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, 256358);