#427296

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6)(7,10)(8,9)(11,12)(13,14,18,23)(15,25,29,40)(16,30,33,21)(17,24,19,22)(20,27,36,28)(26,42,31,41)(32,38,34,39)(35,44,37,43);(1,2)(3,4)(13,15)(14,20)(16,31)(17,27)(18,29)(19,28)(21,37)(22,40)(23,36)(24,25)(26,33)(30,35)(32,44)(34,43)(38,41)(39,42);(2,4)(6,9)(10,12)(13,16,18,33)(14,21,23,30)(15,26,29,31)(17,32,19,34)(20,35,36,37)(22,38,24,39)(25,41,40,42)(27,43,28,44);(5,7,8,11)(6,10,9,12)(14,22)(15,27)(17,19)(20,36)(21,38)(23,24)(26,43)(28,29)(30,39)(31,44)(32,34)(35,37)

Polycyclic presentation:

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

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, 334492);