#7563557

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1722 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=g,d^2=g,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=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,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, 7470753);