#6617451

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25729	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3075 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6524647);