#304278

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=ef,d^2=eh,f^2=h,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^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211474);