#6576505

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#36082	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2789 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483701);