#6993976

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2228 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)(5,6)(7,9)(8,10)(11,12)(13,14,20,24)(15,25,16,22)(17,18,23,19)(21,28,27,26);(5,6)(7,11)(8,10)(9,12)(13,15,20,16)(14,21,24,27)(17,25,23,22)(18,26,19,28);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,20,15)(14,22,24,25)(17,27,23,21)(18,26,19,28);(7,12)(9,11)(13,17)(14,18)(15,21)(16,27)(19,24)(20,23)(22,26)(25,28);(6,10)(9,11)(13,18,20,19)(14,23,24,17)(15,26,16,28)(21,22,27,25)

Polycyclic presentation:

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

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, 6901172);