#7064674

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	#42970	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1852 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:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=h,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

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, 6971870);