#6606682

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#39491	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1892 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513878);