#6439444

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	#23426	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308156, #72308271, #72308312
maximal normal subgroup of	#72308156, #72308271, #72308312

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346640);