#323073

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

804 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,c^2=h,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230269);