#321456

order	512
nilpotency class	3
isoclinism	isoclinic class of #298795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

885 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=egh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228652);