#348625

order	512
nilpotency class	3
isoclinism	isoclinic class of #348602
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	#3482	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=i,d^2=i,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,d^b=dh,f^c=fi,g^c=gi,f^d=fi,g^d=gh

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, 255821);