#202519

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
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

759 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^c=di,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, 109715);