#345559

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

546 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:

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=ei,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

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, 252755);