#441094

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35564	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1576 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)(6,10)(11,12)(13,14,18,26,17,20,16,21)(15,22,25,19,23,28,24,27);(2,3)(5,6,9,10)(7,11,8,12)(15,23)(19,27)(22,28)(24,25);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,15)(14,19)(16,25)(17,23)(18,24)(20,27)(21,28)(22,26);(1,3,4,2)(5,8,9,7)(6,11,10,12)(14,20)(16,18)(19,27)(24,25)

Polycyclic presentation:

2^9|a^2=fg,b^2=egi,c^2=e,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=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, 348290);