#392940

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1850 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)(6,8)(9,10,12,11)(13,14)(15,23)(16,20)(17,21)(18,22)(19,25)(24,28)(26,27);(2,4)(9,11,12,10)(13,15)(14,19)(16,24)(17,25)(18,26)(20,23)(21,27)(22,28);(5,6,7,8)(13,16,18,17)(14,20,22,21)(15,24,26,25)(19,23,28,27);(14,21)(16,17)(19,27)(20,22)(23,28)(24,25)

Polycyclic presentation:

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

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, 300136);