#392853

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1990 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)(6,7)(9,10,12,11)(13,14,16,22)(15,23,24,19)(17,20,18,21)(25,28,26,27);(5,6,8,7)(9,11,12,10)(13,15)(14,19)(16,24)(17,25)(18,26)(20,27)(21,28)(22,23);(2,4)(5,7,8,6)(14,20)(17,18)(19,27)(21,22)(23,28)(25,26);(14,21)(15,24)(17,18)(19,27)(20,22)(23,28)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,d^b=di,f^c=fi,g^c=gi,f^d=fi,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, 300049);