#120771

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#400
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

538 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=d,d^2=i,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,f^b=fhi,g^b=ghi,f^c=fh,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27967);