#7560651

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

2001 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:

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

Polycyclic presentation:

2^9|c^2=fgi,d^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7467847);