#7562146

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#44884	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2216 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fgi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7469342);