#471584

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1189 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:

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

Polycyclic presentation:

2^9|b^2=egh,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de,f^c=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378780);