#355175

order	512
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1216 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)(7,11)(13,14,19,22,18,23,17,26)(15,21,24,27,16,20,25,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,18)(17,25)(19,24)(21,23)(22,28)(26,27);(6,9)(10,12)(13,16,18,15)(14,21,23,20)(17,25,19,24)(22,28,26,27);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,22,23,26)(15,24,16,25)(20,27,21,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262371);