#356823

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂×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 ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1102 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=de,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264019);