#7564169

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1937 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|c^2=fgi,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=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, 7471365);