#7551204

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	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2109 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)(8,9)(10,12)(13,14,19,23)(15,24,22,18)(16,20,21,28)(17,26,27,25);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,14,19,22,24,23)(16,17,25,20,21,27,26,28);(6,9)(10,11)(13,16,19,21)(14,20,23,28)(15,17,22,27)(18,25,24,26);(2,4)(5,7,8,12)(6,10,9,11)(16,21)(17,27)(20,28)(25,26);(13,17,18,20,19,27,24,28)(14,21,22,26,23,16,15,25)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,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, 7458400);