#315554

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=efi,c^2=hi,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222750);