#7503957

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500908
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#31071	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1754 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,4)(5,7)(6,8)(9,10,14,20,15,19,34,39)(11,16,24,35,23,27,25,36)(12,22,28,37,29,40,33,17)(13,21,30,18,32,38,31,26);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,26)(13,22)(14,24)(15,23)(17,31)(18,29)(19,27)(20,35)(21,28)(25,34)(30,37)(32,40)(33,38)(36,39);(9,12,15,29)(10,17,19,37)(11,21,23,38)(13,27,32,16)(14,33,34,28)(18,24,26,25)(20,40,39,22)(30,35,31,36);(2,7)(3,8)(9,13)(10,18)(11,22)(12,27)(14,31)(15,32)(16,29)(17,24)(19,26)(20,21)(23,40)(25,37)(28,35)(30,34)(33,36)(38,39);(1,4,5,6)(2,3,7,8)(10,19)(11,23)(17,37)(18,26)(20,39)(21,38)(22,40)(24,25)

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411153);