#121416

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#385
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

680 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28612);