#6502932

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
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	#37272	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2061 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6410128);