#354260

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2372 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)(3,4)(5,6,8,7)(13,14)(15,23)(16,21)(17,20)(18,22)(19,25)(24,27)(26,28);(2,4)(9,10,11,12)(13,15)(14,19)(16,26)(17,25)(18,24)(20,27)(21,23)(22,28);(14,20)(16,17)(19,27)(21,22)(23,28)(25,26);(5,7,8,6)(15,24)(19,28)(23,27)(25,26)

Polycyclic presentation:

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

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, 261456);