#7551003

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#24440	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2221 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:

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

Polycyclic presentation:

2^9|b^2=f,d^2=gi,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7458199);