#355373

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^b=di,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262569);