#6683494

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2436 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)(6,9)(7,11)(13,14,19,23)(15,17,25,20)(16,26,28,21)(18,22,24,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,21)(17,23)(18,27)(19,25)(22,24)(26,28);(1,2,3,4)(13,16,19,28)(14,21,23,26)(15,24,25,18)(17,22,20,27);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,15,23,25)(16,27,28,22)(18,26,24,21);(6,9)(10,12)(13,18)(14,22)(15,16)(17,26)(19,24)(20,21)(23,27)(25,28)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=f,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	#72257411, #72257431, #72264241
maximal normal subgroup of	#72257411, #72257431, #72264241
maximal quotient of	#72308555

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590690);