#123522

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,g^b=gi,g^c=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, 30718);