#7149286

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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2022 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,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, 7056482);