#7533522

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#2541	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1487 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7440718);