#7521936

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7429132);