#357198

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1476 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=egi,d^2=ghi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=ei,g^b=gi,d^c=di

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, 264394);