#6578871

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#35752	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2033 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)(5,6)(7,11)(8,10)(9,12)(13,14,18,23)(15,25,27,20)(16,21,19,24)(17,22,28,26);(5,7,9,8)(6,10,12,11)(13,14)(15,26)(16,24)(17,20)(18,23)(19,21)(22,27)(25,28);(2,4)(5,7,9,8)(6,11,12,10)(13,15,18,27)(14,20,23,25)(16,17,19,28)(21,26,24,22);(5,7,9,8)(6,11,12,10)(13,16)(14,21)(17,28)(18,19)(22,26)(23,24);(5,8,9,7)(6,10,12,11)(13,17)(14,22)(15,19)(16,27)(18,28)(20,21)(23,26)(24,25)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307072, #72307285, #72307607
maximal normal subgroup of	#72307072, #72307285, #72307607

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486067);