#348054

order	512
nilpotency class	3
isoclinism	isoclinic class of #348038
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18335	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255250);