#448831

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,6)(3,5,7,8)(17,18,23,24,22,19,20,26)(21,25,28,29,31,27,30,32);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,26,23,24)(21,29,31,32)(25,30,27,28);(2,6)(5,8)(10,13)(14,16)(17,20,22,23)(18,24,19,26)(21,28,31,30)(25,32,27,29);(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,30)(24,32)(26,29)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=i,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei

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, 356027);