#321495

order	512
nilpotency class	3
isoclinism	isoclinic class of #298795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

899 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)(5,6,8,11,7,9,12,10)(13,14)(15,20)(16,25)(17,22)(18,21)(19,24)(23,27)(26,28);(1,2,4,3)(6,9)(10,11)(13,15,16,26,17,24,18,23)(14,19,25,27,22,20,21,28);(6,10)(8,12)(9,11)(13,15,17,24)(14,20,22,19)(16,23,18,26)(21,28,25,27);(1,3,4,2)(6,9)(10,11)(13,16,17,18)(14,21,22,25)(15,23,24,26)(19,27,20,28)

Polycyclic presentation:

2^9|a^2=f,b^2=egh,c^2=i,d^2=eh,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, 228691);