#7557547

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#35736	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1825 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,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, 7464743);