#6583969

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2324 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,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	#72257270, #72257349, #72257612, …
maximal normal subgroup of	#72257270, #72257349, #72257612, …
maximal quotient of	#72265034, #72265055, #72280860

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491165);