#224118

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(2,5)(6,7)(10,13)(11,15)(14,16)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,2,4,5)(3,7,8,6)(10,14)(13,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(9,10)(11,16)(12,14)(13,15)(19,25)(22,28)(26,31)(29,32);(1,3,4,8)(2,6,5,7)(19,25)(22,28)(26,31)(29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=chi,d^b=dei,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 131314);