#7003204

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

2018 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)(5,6)(7,9)(8,10)(11,12)(14,19)(16,17)(20,27)(21,22)(23,28)(25,26);(5,6)(7,11)(8,10)(9,12)(13,14,16,21,18,22,17,19)(15,23,26,27,24,20,25,28);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,24)(14,20,22,23)(16,26,17,25)(19,27,21,28);(7,12)(9,11)(14,21)(15,24)(16,17)(19,22)(20,27)(23,28);(6,10)(9,11)(13,16,18,17)(14,19,22,21)(15,25,24,26)(20,28,23,27)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,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,d^c=dh,e^c=ei,e^d=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, 6910400);