#7562184

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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	#59003	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2383 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)(7,11)(8,12)(9,10)(13,14)(15,21)(16,27)(17,28)(18,23)(19,24)(20,25)(22,26);(1,2)(3,4)(6,10)(11,12)(13,15)(14,21)(16,19)(17,18)(20,26)(22,25)(23,28)(24,27);(5,7,9,8)(6,11,10,12)(13,16,18,26,20,17,19,15)(14,22,24,27,25,21,23,28);(2,4)(5,8,9,7)(6,12,10,11)(13,17,18,15,20,16,19,26)(14,21,24,28,25,22,23,27);(6,10)(11,12)(13,18,20,19)(14,23,25,24)(15,17,26,16)(21,28,22,27)

Polycyclic presentation:

2^9|c^2=fg,d^2=fg,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^c=ei,e^d=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, 7469380);