#6670198

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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

2130 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)(6,9)(7,11)(13,14,18,21)(15,22,25,17)(16,27,19,20)(23,26,28,24);(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,20)(17,26)(18,21)(19,27)(22,24)(25,28);(2,4)(13,15)(14,17)(16,26)(18,25)(19,24)(20,28)(21,22)(23,27);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,20,21,27)(15,24,25,26)(17,23,22,28);(6,9)(10,12)(13,17)(14,15)(16,23)(18,22)(19,28)(20,24)(21,25)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257411, #72257444, #72264140
maximal normal subgroup of	#72257411, #72257444, #72264140
maximal quotient of	#72265680

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6577394);