#229814

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	#2011	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

762 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)(5,6)(7,9)(8,11)(10,12)(13,14,18,22,17,21,16,20)(15,19,23,26,24,27,25,28);(2,3)(13,14,16,20,17,21,18,22)(15,19,25,28,24,27,23,26);(6,9)(7,12)(10,11)(13,15)(14,19)(16,25)(17,24)(18,23)(20,28)(21,27)(22,26);(1,3,4,2)(13,16,17,18)(14,20,21,22)(15,23,24,25)(19,26,27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dei,d^c=di,f^c=fh

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, 137010);