#6667568

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2920 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)(7,11)(8,10)(9,12)(13,14,16,19)(15,21,24,18)(17,20,25,27)(22,26,28,23);(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,24)(17,26)(19,21)(20,22)(23,25)(27,28);(2,4)(13,14)(15,22)(16,19)(17,27)(18,26)(20,25)(21,23)(24,28);(6,10)(9,11)(14,19)(18,21)(20,27)(22,28);(7,12)(9,11)(15,23)(17,25)(18,28)(20,27)(21,22)(24,26)

Polycyclic presentation:

2^9|a^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257583, #72264133, #72264223
maximal normal subgroup of	#72257583, #72264133, #72264223
maximal quotient of	#72265071

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6574764);