#120613

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#389
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,f^b=fi,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27809);