#348927

order	512
nilpotency class	3
isoclinism	isoclinic class of #346644
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

931 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,d^b=di,f^c=fh,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256123);