#6996603

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#44892	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2376 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6903799);