#359148

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1600 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ei,d^2=hi,f^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,d^c=di,f^c=fi

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, 266344);