#304766

order	512
nilpotency class	3
isoclinism	isoclinic class of #271643
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

762 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=e,c^2=efh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=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, 211962);