#149040

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=d,d^2=hi,f^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fhi,g^b=gh,h^b=hi

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, 56236);