#7063042

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994107
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 ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2081 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)(13,14,18,22)(15,20,24,27)(16,21,17,19)(23,26,25,28);(5,6)(7,11)(8,10)(9,12)(14,19)(15,23)(16,17)(21,22)(24,25)(26,28);(2,4)(5,7,8,12)(6,9,10,11)(15,24)(20,27)(23,25)(26,28);(7,12)(9,11)(13,15,18,24)(14,20,22,27)(16,25,17,23)(19,26,21,28);(6,10)(9,11)(13,16,18,17)(14,21,22,19)(15,25,24,23)(20,28,27,26)

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^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=dfi,e^b=ehi,f^b=fi,d^c=dhi,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, 6970238);