#7148365

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
prime radical	#25602	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1892 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)(7,11)(10,12)(13,14,18,22)(15,20,17,19)(16,21,24,28)(23,27,25,26);(6,9)(7,11)(14,19)(15,17)(16,23)(20,22)(21,28)(24,25);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,17)(14,20,22,19)(16,23,24,25)(21,27,28,26);(6,9)(10,12)(13,16,18,24)(14,21,22,28)(15,23,17,25)(19,26,20,27);(5,7,8,11)(6,10,9,12)(13,15,18,17)(14,20,22,19)(16,23,24,25)(21,27,28,26)

Polycyclic presentation:

2^9|a^2=gi,c^2=i,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,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, 7055561);