#7522576

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

2356 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)(8,10)(9,12)(13,14)(15,23)(16,21)(17,19)(18,22)(20,24)(25,27)(26,28);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,14)(15,23)(16,21)(17,19)(18,22)(20,24)(25,27)(26,28);(14,19)(16,17)(20,23)(21,22)(24,26)(27,28);(2,4)(5,7,9,8)(6,11,12,10)(13,15)(14,20)(16,24)(17,26)(18,25)(19,23)(21,27)(22,28);(5,8,9,7)(6,11,12,10)(13,16,18,17)(14,21,22,19)(15,24,25,26)(20,27,28,23)

Polycyclic presentation:

2^9|b^2=f,d^2=f,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,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, 7429772);