#6577477

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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2066 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)(14,20)(15,22)(21,26)(24,27);(5,7,9,8)(6,10,12,11)(13,14)(15,23)(16,21)(17,24)(18,20)(19,26)(22,28)(25,27);(2,4)(5,7,9,8)(6,11,12,10)(13,15,18,22)(14,17,20,25)(16,24,19,27)(21,28,26,23);(5,8,9,7)(6,10,12,11)(13,16)(14,21)(15,22)(18,19)(20,26)(23,28);(13,17,18,25)(14,15,20,22)(16,23,19,28)(21,27,26,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307072, #72307127, #72307472
maximal normal subgroup of	#72307072, #72307127, #72307472

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6484673);