#7566073

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#44892	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7473269);