#7149369

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	C₂×#3224	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1963 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)(15,18)(16,24)(17,19)(20,22)(21,27)(23,26)(25,28);(6,9)(7,11)(13,15,18,14,20,17,19,22)(16,25,28,24,27,23,26,21);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,23)(17,25)(18,28)(19,26)(20,27)(22,24);(6,9)(10,12)(13,17,20,15)(14,19,22,18)(16,23,27,25)(21,28,24,26);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,15,22,17)(16,26,27,28)(21,25,24,23)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,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, 7056565);