#348030

order	512
nilpotency class	3
isoclinism	isoclinic class of #348018
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	#18335	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

881 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)(19,26)(20,23)(25,30)(27,28)(29,31);(2,6)(7,15)(8,13)(9,16)(17,18,23,30,21,25,20,22)(19,24,26,31,28,32,27,29);(1,3)(2,7)(4,10)(5,14)(6,15)(8,16)(9,13)(11,12)(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,32)(30,31);(5,12)(8,13)(9,16)(11,14)

Polycyclic presentation:

2^9|b^2=ei,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,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, 255226);