#6576160

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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	#24364	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2071 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)(5,6,8,9)(7,10,11,12)(14,20)(15,24)(22,27)(23,26);(6,9)(10,12)(13,14)(15,25)(16,22)(17,23)(18,20)(19,26)(21,27)(24,28);(2,4)(13,15,18,24)(14,21,20,16)(17,27,19,22)(23,28,26,25);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,26)(17,28)(18,21)(19,25)(20,27)(23,24);(13,17)(14,20)(15,22)(18,19)(24,27)(25,28)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,e^c=ei,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	#72307099, #72307249, #72307478
maximal normal subgroup of	#72307099, #72307249, #72307478

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483356);