#279238

order	512
nilpotency class	3
isoclinism	isoclinic class of #279211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3038	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

541 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,g^b=gh,d^c=dh,f^c=fi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 186434);