#6575767

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2198 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)(7,10)(8,9)(11,12)(13,14,18,23)(15,20,24,28)(16,22,17,21)(19,26,27,25);(1,2)(3,4)(6,9)(10,12)(15,24)(19,27)(20,28)(25,26);(2,4)(13,15,18,24)(14,19,23,27)(16,25,17,26)(20,22,28,21);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,26)(17,25)(18,24)(19,21)(22,27)(23,28);(2,4)(13,16,18,17)(14,21,23,22)(15,25,24,26)(19,20,27,28)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482963);