#354157

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^b=di,f^c=fi,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261353);