#6606408

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	#38049	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1810 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)(17,21)(18,22)(19,28)(20,23)(24,26)(25,27);(2,4)(5,6,8,9)(7,10,11,12)(13,15,20,25)(14,16,23,27)(17,19,24,18)(21,28,26,22);(6,9)(7,11)(13,16,18,26,20,27,19,21)(14,17,28,25,23,24,22,15);(5,7)(6,10)(8,11)(9,12)(13,17,20,24)(14,21,23,26)(15,18,25,19)(16,22,27,28);(6,9)(7,11)(13,18,20,19)(14,22,23,28)(15,24,25,17)(16,26,27,21)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513604);