#511536

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=ef,d^2=efi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343697, #72343811, #72343826
maximal normal subgroup of	#72343697, #72343811, #72343826

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418732);