#6576169

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	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2041 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)(14,19)(20,27)(21,28)(22,26);(6,9)(10,12)(13,14,17,19)(15,21,25,28)(16,20,23,27)(18,26,24,22);(2,4)(13,15,17,25)(14,20,19,27)(16,18,23,24)(21,22,28,26);(5,7,8,11)(6,10,9,12)(15,23)(16,25)(18,24)(20,28)(21,27)(22,26);(13,16,17,23)(14,21,19,28)(15,24,25,18)(20,26,27,22)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=fi,e^2=i,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, #72307267, #72307281
maximal normal subgroup of	#72307099, #72307267, #72307281

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483365);