#202523

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

715 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dh,d^c=di,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109719);