#381838

order	512
nilpotency class	3
isoclinism	isoclinic class of #29751
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

852 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=fh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=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, 289034);