#7528296

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2433	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1639 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,30)(28,32)(29,31);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,20,27)(18,22,23,25)(21,26,29,28)(24,32,31,30);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(18,23)(22,25)(24,31)(30,32);(10,14)(13,16)(18,23)(19,26)(21,29)(22,30)(25,32)(27,28);(1,4,5,3)(2,7,6,8)(18,23)(19,27)(24,31)(26,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,h^b=hi,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, 7435492);