#321771

order	512
nilpotency class	3
isoclinism	isoclinic class of #278923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#28450	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

873 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eghi,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^c=fi

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, 228967);